Talk:Multilateration

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[edit] Math errors

I think that the math in the basic formula is a bit confused. (x, y, z) is given as the unknown location of the emitter, but then also as the location of the central transmitter. Different variable names should be used.

While I mention this, I would also suggest that the mathematical discussion be fleshed out a bit more by listing what should be minimized in the optimization.

Mark (talk) 19:46, 29 April 2008 (UTC)

[edit] Decca

I am no expert in this field. But I would like clarify a small matter.

My understanding of the situation (and its 20 years since I did my exams) is that the LORAN hyperbolae were based on time differences, but that Decca and Omega were both based on phase differences and NOT time differences.

If this is the case then one of the following actions should be taken:

1. The reference to multilateration should be removed from the decca article; or 2. The definition on the multilateration page should be changed to include phase or time difference


[edit] Response

I have modified the article to include mention of the phase approach, and also modified the DECCA article to say "approach similar to multilateration". In fact, as the DECCA transmissions are continous wave, I think it is correct to describe the approach as multilateration. Phase-difference and time-difference are essentially the same thing with a narrow band source. Paul 06:42, 26 December 2005 (UTC)


This does not make sense to me. Either Multilateration includes phase difference, in which case the description "approach similar to multilateration" is incorrect for Decca, or it excludes phase difference, in which case Decca is not multilateration. The NPL describes it as "Multi-lateration is a way of measuring the position of a target, or targets, relative to several fixed measuring stations" at http://www.npl.co.uk/length/dmet/science/multilateration.html. This appears to include phase difference. --SC 22:17, 29 December 2005 (UTC)
I think that the confusion originates from the fact that this entry describes Multilateration as "the process of locating an object by accurately computing the time difference of arrival of signals from three or more locations to that point". Having read around a bit more it appears to me that Multilateration measurements do not have to be by TDOA (although they presumably are usually). This article also duplicates the principles already covered in the trilateration article. I propose the following actions:
  • Rewrite multilateration removing dependence on TDOAs, whilst recognising that TDOAs are often used.
  • Remove duplication between multilateration and trilateration. (The Trilateration article is probably a better place to describe the principles, since it is an easier concept to grasp.)
  • Reduce emphasis on Decca in both articles, since it is obsolete and (I think) that GPS is hyperbolic also, so that would be a better reference
  • Add description of what multilateration adds to trilateration. This is unclear to me.

--SC 08:19, 30 December 2005 (UTC)


The problem with using GPS as an example is that it is very difficult to plot GPS hyperbolae on a chart or map. For a start they are 3-dimensional. They are time-difference based and the stations are in orbit above us.
The nice thing about Decca and/or Loran is that the hyperbolae can be (and often are) overlayed on a navigational chart and the readings plotted as lines of position to give a position fix. It was certainly seeing overlayed hyperbolae printed on charts that made it all "click" for me all those years ago. I seem to remember our Electronic Nav lecturer using a North Sea chart with the lattices from 3 different Decca chains printed thereon and the positions of at least 2 master stations and numberous slave stations highlighted along with the baselines. It just all fell into place.
Now trying to do something similar with GPS LOP's, that would be fun.Frelke 09:44, 30 December 2005 (UTC)


Again I am not claiming expert status here, but isn't multilateration just a generic form of trilateration, i.e tri = 3 and multi = >1. I am guessing now, but I think we are headed for merge here.

Frelke 09:49, 30 December 2005 (UTC)

[edit] Clarification

I'm struggling to understand the confusions here. Multilateration is the determination of location using multiple receivers. Trilateration is with exactly three. Both use TDOA to determine the intersection of 2 (with trilateration) or N-1 (with multilateration using N receivers) hyperboloids. Hence the term hyperbolic positioning.

TDOA is usually measured by measuring time of arrival directly, but equivalently can be determined by measuring phase difference - but only if the signal is narrowband.

If anything needs to be merged, then trilateration should be merged into this article, as trilateration is just a special case of multilateration.

The article already says all of this. Paul 21:37, 18 January 2006 (UTC)

[edit] It's mixed up

Quote «Multilateration, also known as hyperbolic positioning...»

1. Multilateration (including trilateration) is based on estimation of the time of arrival (TOA).
2. Hyperbolic positioning is based on estimating the time difference of arrial (TDOA).
3. Dopler positioning is based on estimating the dopler shift of the satellite signal.

These are three (3) main (and different) types of radionavigation. See, for example, book Global Positioning System by Pratap Misra and Per Enge (page 12, chapter 1.2 "Methods of Radionavigation").


Kender 05:37, 18 January 2006 (UTC) Stanford, CA

Now I'm totally confused. So what is the difference between types 1 and 2 above ?
Is there a difference between is based on estimation of the time of arrival and is based on estimating the time difference of arrial. I think we have already agreed here that Hyperbolic positioning can be based on more than just time difference. It can be based on phase difference (as in the case of Decca and Omega). So can we clarify what Misra and Enge mean (presumably these definitions are theirs).
My biggest problem with all this is that I still believe that whatever multilateration is, the only difference between it and trilateration is the number of position lines used, tri having 3 p/l's and multi having an unspecified number.
Frelke 07:35, 18 January 2006 (UTC)


It's all based on the difference between TDOA and TOA.

Since historically TDOA was used for navigation before TOA, let's start with TDOA. Consider two transmitters spaced far apart and one receiver (or user). Each of the transmitters sends a pulse at the same time – they are synchronized. A user first receives a pulse from transmitter 1 then from transmitter 2. The delay between the pulsed is TDOA. TDOA=TOA1-TOA2 (Because of the clock bias, user doesn’t even know the TOAs.) In 2D TDOA from one pair of transmitters puts a user on the hyperbola; hence TDOA systems are also known as hyperbolic systems. To estimate the position the user needs at least two pairs of transmitters (two TDOAs). Each of the pairs will produce a hyperbola and the user position is at the intersection of these hyperbolas.

Next, consider one transmitter and one receiver. One pulse arrives to the receiver. There is no TDOA, because you need two pulses to produce the difference. However, if both receiver and transmitter are somehow synchronized to common time, TOA can be estimated. In 2D TOA from one receiver puts a user on a circle. To estimate the position the user needs at least two transmitters (two TOAs).

Quote «The problem with using GPS as an example is that it is very difficult to plot GPS hyperbolae on a chart or map.»

GPS doesn't produce hyperbolas, because it's not a TDOA system. It's a TOA systen, and the LOP in 3D is a sphere.

Kender 08:17, 18 January 2006 (UTC) Stanford, CA


[edit] the above is mixed up!

The statement "multilateration is based on estimation of the time of arrival (TOA)" above is incorrect. Multilateration uses the time-difference of arrival of a pulse between two sites (see, for instance, [1] or [2]). Absolute time of arrival is not required, and not even measured in systems such as VERA. The explanation above of two transmitters and one receiver is correct - but just the reciprocal case of what I just described. Both are TDOA. So, in terms of the list above, both 1) and 2) use TDOA, and both can be called multilateration or hyperbolic positioning. Incidentally, in my professional life I work on this technology, and the term multilateration is commonly used in the way described in this article.

Paul 06:16, 19 January 2006 (UTC)

[edit] Merge

I am proposing to merge this page with trilateration, this being the more general case. I think that the other article is actually the better article and so would intend to keep the vast majority of it.

See discussion page for vote.

Frelke 07:42, 19 January 2006 (UTC)

I would agree - but see my more detailed comments on discussion page. Care is required. Paul 11:32, 19 January 2006 (UTC)
Frelke, do you think that trilateration is a particular case of multilateration? Kender 04:15, 23 January 2006 (UTC) Stanford, CA
I think that it is, but I am not an expert. I just studied hyperbolic navigation many years ago and my brain still works. Frelke 11:18, 23 January 2006 (UTC)

[edit] Loose ends?

The article is much improved after the recent activity. I presume it is correct now ;-) If so, there appear to be a few loose ends to tidy up as the Multilateration article says that Decca used Multilateration but the Decca article says that is used "an approach similar to multilateration". Does this mean now that the Decca article (and others) needs updating? --SC 22:43, 27 January 2006 (UTC)

Again, I am not the expert, rather the person who stirs the bucket to get a reaction. I remember that comment going into the Decca article and thinking myself at the time that it sounded 'uncomfortable'.
I suggest that it be changed to "...a multilateration-based approach..." unless anyone has a better alternative. Frelke 23:25, 27 January 2006 (UTC)
I just updated the DECCA article seeing as it was my unfortunate wording in the first place. I simply changed it to "also known as multilateration". --Paul 12:29, 30 January 2006 (UTC)


[edit] GPS

GPS works by receiving several known signals in 1 location, not by sending one signal, and receiving in several locations. So I'd say that GPS is not a valid example (but location determining in cellphone networks can be). Am I right?

I would rather say that where you do the measurement is of no important. The basic principle is the same. You measure the difference in distances between several references and an unknown position. If the references or the "locator" sends the signals is irrelevant. (Håkon K. Olafsen) 193.157.188.206 14:31, 25 March 2007 (UTC)

[edit] Three receivers - one additional TDOA?

I think the Principle section should be reformulated in this way:

For two receivers we get one hyperboloid (either one of the pair, depending on the sign of TDOA). Adding THIRD receiver brings in TWO additional hyperboloid, because for each pair of receivers there is one (assuming the three receivers are not on one line). So two of these hyperboloids give us a curve while the third one should pin-point the location precisely. Am I right?
MiShogun 08:59, 31 July 2007 (UTC)
For two receivers you get one hyperboloid. Because you have one TDOA. Adding a THIRD receiver gives you three TDOA, but they are dependend on each other. TDOA3 = TDOA1 + TDOA2. Because they are dependend you still have actually only two nonrelated parameters. To be able to describe any point in space you need at least 3 nonrelated parameters. The TDOA3 does not give any extra information. The first two paraboloids intersect in a closed curve the third intersects the same curve as wel. You need a Fourth receiver to be able to pin-point. And then there are still 2 solutions, two pin points.
Or for GPS you need the reception of 4 satellites to do 3D pinpointing.
Crazy Software Productions 17:32, 5 September 2007 (UTC)
You're almost right. With 4 reference points and known difference in distance between the references and the unknown position, it is possible to find one unique solution in space (3D). You can also see Trilateration for this.
With 4 reference points (4 receive times, so 3 independend differences), there are two solutions (x1,y1,z1,t1) and (x2,y2,z2,t2). The second point is not stable, moving fast and the solution can have a 'large' time offset from the actual time. So although there are two solutions, it's easy to bin one of the solutions. But somewhere in space and somewhere in time there is a point with exactly the same time differences as the 'actual' point. The algoritm show at trilateration starts from knowing distances and at the start of the calculation they are not available..Crazy Software Productions 14:38, 8 September 2007 (UTC)
It's obviously something I'm not getting here. Where does t1 and t2 come from? With four reference points, there are one unique solution to a localisation problem in 3D (as long as the references are not in a line, in the same spot and a few other special cases).
With four reference points there are two solutions. With almost all configurations there are two solutions. One of the solution is not stable, far removed from earth and with a time-error which can be huge, but there is a second solutions for the four reference points. If you are working with the model with the spheres, you normally alter the size of the spheres with the same amount until they intersect at one point. But if you keep changing the size of the spheres (often to very large) they will intersect in another point, often far removed from the first point and with spheres of a larger size. (The size of the spheres represent time timing (error)). —Preceding unsigned comment added by Crazy Software Productions (talk • contribs) 13:44, 18 September 2007 (UTC)
Multilateration as I've understood and used it, does not necessary involve TDOA measurements. It's the mathematical calculation of the localisation when the difference in distance (time transfers to distance) from the unknown position to the references are known. I think this article needs to be completely rewritten, and multilateration treated as a mathematical concept, and be distinguished from TDOA measurements.
GPS with a quartz clock is based on TDOA measurements, just because no absolute time is available. I try to avoid the terms Trilateration and Multilateration because they are not completely clair. TOA and TDOA are better defined. I would think that trilateration (as described in a lot of places) is TOA put at some places this is contested. TDOA gives hyperbolic shapes (althought this does not have to present itself in the calculation) and that this is often refered to as Multilateration.
My point is that trilateration and multilateration is two techniques used to solve a localisation problem when you got a set of data. The data used in multilateration is basically the difference in distance to certain reference points. You do not necessary find this difference with TDOA. If you measure the distances to the references, multilateration can ease the calculations and help improve the accuracy by suppressing errors in the measurements.
The time / distance measurements does not even have to be synchronous, as long as the unknown position is not moving, or moving very slowly.
If the time /distance measurements are not done very close together in time, then the accuracy of the clock becomes very important. With a very good clock, you could do position calculation with only one satellite (in four different places), you only need four measurements, but a extremely accurate clock is needed. (The drift in a very good quartz clock is 10 meters each second. The drift in a 'normal' quartz clock is 500 meters each second. So if the measurements are done within a second, with a normal clock, the position calculation is not very accurate anymore).Crazy Software Productions 14:38, 8 September 2007 (UTC)
This is one of the problems by having a to close relationship with TDOA and multilateration. You can used the same equations as multilateration, except the 1/c part if you know the distances. TDOA needs multilateration, multilateration does not need TDOA.
Haakoo 05:23, 17 September 2007 (UTC)
Haakoo 06:10, 7 September 2007 (UTC)
Thanks for clearing this up. Took me a long time to get what you are saying. As I get it now for multilateration there could be other methods than time differencing. (At the moment I can think of none). Part of the confusion I think comes from that so many people try to explain tri and multi as confined to certain numbers. I think that the tri is from triangle, and that three points form a triangle so for triangulation two known points can define the third point, equivalent for trilateration again based on triangles. And not that three known points should be given. Multilateration for me at least suggests a number (maybe more than tri?), but could also mean multi triangles, or multiple sources for one parameter (difference of two signals/distances ?). So it took me quite some time to get you were talking that TDOA also defines how we get the info and multilateration just does not specify how we get the info. I don't know any application of multilateration which is not based on TDOA, so I thought of them as equivalent (or almost the same). (Sorry My mistake.).
Do you have an example of multilateration which is not based on TDOA?
Crazy Software Productions 18:17, 18 September 2007 (UTC)
The name is confusing, and I had a good time trying to track down different localisation algorithms when I was working on my master thesis. I'm not sure why they are called multilateration and trilateration, might be because the equations used in multilateration are easily used in an over determined system. If you take a look at my master thesis I (we) consider WSNs where the nodes can measure the distance between themselves, and then creates a grid of locations from this information. The technique used to estimate the distance (range) between nodes is found in my friends master thesis. It is not a working system, but you get the idea.
My master thesis "Wireless Sensor Network Localisation Strategies" is available from: http://wo.uio.no/as/WebObjects/theses.woa/wa/these?WORKID=60422 Chapter 4.7.3-4 show the mathematics for "multilateration" and an over determined case. And chapter 3.4 discuss data acquisition.
Nikolaj's master thesis "High Precision Ranging in Wireless Sensor Networks" is available from: http://wo.uio.no/as/WebObjects/theses.woa/wa/these?WORKID=58956 Chapter 3 is probably most interesting.
I guess my biggest problem with the multilateration article is that Hyperbolic positioning and TDOA points diretcly to multilateration, and if this is to continue, multilateration needs to be rewritten. Or we can have the mathematical calculations and explenations in Hyperbolic positioning and let multilateration be (HP+)TDOA. I'm not sure what's the right thing to do.
Haakoo 07:01, 22 September 2007 (UTC)


[edit] Major clean up

This article with it's redirects really needs a major clean up. The Multilateration article should either be similar to the Trilateration article, without the strong connection to Time difference of arrival / TDOA or Hyperbolic positioning should not redirect here. I do not have any good references for the present norm in literature. My opinion is that the Trilateration article is better than this one, causing less confusion. It basically ends up with a discussion of the meaning of "Multilateration", and how it should be used.

If multilateration is considered to be the process of doing a TDOA and the estimate the position, Hyperbolic positioning should not be redirected here. If multilateration on the other hand is the process of estimating a position based on a given set of data (difference in distance to reference points of known position) then TDOA should not redirect here and the article rewritten.

TODA deserves it's own article just as Time of arrival, and this article should be rewritten without the strong connection to TDOA. As I've said earlier, multilateration does not need TDOA data to estimate a position.

Haakoo 02:56, 26 September 2007 (UTC)

[edit] Terrestrial radionavigation

The article describes 3D-space case, where distances are straight lines, but no words are said about spherical case of terrestrial radionavigation (e.g. Loran-C), where distances are calculated using haversine formula. Unomano 07:06, 11 October 2007 (UTC)