Talk:Trilateration

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"Trilateration is most commonly used in GPS applications, where the distances from satellite transmitters (the reference points) are measured by a receiver (the subject point)."

Not precisely. What are measured are timing differences, so we are really intersecting hyperboloids of revolution. It would be trilateration if we already knew the exact time; but that's one of the things we are solving for. -- The Anome 11:59, 27 Aug 2004 (UTC)

Contents

[edit] locations at sea

For example, if the subject is known to be on land, or on the surface of the Earth, and one of the candidate locations is at sea or in space, that point may be disregarded.

Why the locations at sea are disregarded? A subject can be at sea or land at any given time.--Assu 14:01, 29 Dec 2004 (UTC)

At any time subject knows where he is. If one of the candidate points is on land and the other one is at sea, and the subject is on land - he can easily pick the correct point from the two. It works the other way as well. Furthermore, if the subject knows his approximate position from previous oservations, he can make a pick as well. Kender 00:46, 25 January 2006 (UTC)

[edit] C++ to do 2D trilateration

Just hand-derived this. It seems to work and might be useful to add to the article.

// Precondition: one center point (p1/p2/p3) must not be colinear with the other two.
Point Trilaterate(const Point &p1, float d1, const Point &p2, float d2, const Point &p3, float d3)
{
        double i1=p1.x,i2=p2.x,i3=p3.x;
        double j1=p1.y,j2=p2.y,j3=p3.y;
        double x,y;

        x = (((2*j3-2*j2)*((d1*d1-d2*d2)+(i2*i2-i1*i1)+(j2*j2-j1*j1)) - (2*j2-2*j1)*((d2*d2-d3*d3)+(i3*i3-i2*i2)+(j3*j3-j2*j2)))/
                ((2*i2-2*i3)*(2*j2-2*j1)-(2*i1-2*i2)*(2*j3-2*j2)));
        y = ((d1*d1-d2*d2)+(i2*i2-i1*i1)+(j2*j2-j1*j1)+x*(2*i1-2*i2))/(2*j2-2*j1);

        return Point(x,y);
}

[edit] Math error?

I'm no math wizard, but shouldn't the last term of the equation for y be:

\frac{(r_1^2-r_2^2+d^2)^2}{8d^2j}

The equation given didn't spit back the expected results for me, but using 8 instead of 4 in the denominator yields the expected result. I'm not confident enough in my algebra to change the actual article based on just my conclusions, so if someone else checks my work and finds it to be right please fix this on the main article.

[edit] Merge

Note: following all the discussion below, I have clarified both Trilateration and Multilateration articles to hopefully address the points below and remove the need for any merging. If the original proposer for the merge, Frelke, is happy with the changes, I suggest the merge tag is removed from the articles --Paul 18:48, 25 January 2006 (UTC)

Just done it. Are we all happy ? Frelke 20:45, 25 January 2006 (UTC)
Yep! It doesn't seem to be confusing any more. Kender 03:47, 26 January 2006 (UTC)
Phew! --Paul 11:32, 26 January 2006 (UTC)

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

[edit] Agree

  1. Strong agree Frelke 07:40, 19 January 2006 (UTC)
  2. I agree but I'm unsure whether to move it to multilateration or keep it at trilateration. The latter is more familiar to laymen while the former is more used in research papers and such. Deco 08:27, 19 January 2006 (UTC)
  3. Agree. This will clarify discussion of the remainer of the issues. I don't know which one should be the main article, but do note for interest that the Encyclopædia Britannica has an article for trilateration, but not for multilateration. --SC 21:47, 19 January 2006 (UTC)

[edit] Disagree

  1. Following the discussions in this section, I now change my vote to disagree as it seems the two concepts are better kept distinct, although I do think we should keep a "See also" in each article, linking to the other. --Paul 12:27, 25 January 2006 (UTC)
  2. I did agree at some point. I've changed my mind. But please stop the confusion. Somebody draws a picture for the TOA positioning, which is rightfully called trilateration, but then talks about the hyperbolic systems and intersection of two hyperboloids. We have to figure out whether tri- and multi-lateration and hyperbolic positioning are the same things. Which ones are TOA and which ones are TDOA? Who and why have coined the term "multilateration", and what sense does it make at all? Any takers? Kender 07:19, 23 January 2006 (UTC) Stanford, CA
See my comments below in "Multilateration vs. Triangulation" Paul 11:47, 23 January 2006 (UTC)

[edit] Discussion on Multilateration vs. Trilateration

It's probably worth trying to come to some agreed position on the relationship between multilateration and trilateration before merging the two articles. To that end, I suggest we use this section. Here is my input for a start! I think points of initial agreement are that "trilateration" normally means using TOA and three sites; similarly "multilateration" normally means three or more sites using TDOA. Also, hyperbolic positioning clearly refers to the use of TDOA and is ambiguous regarding the number of sites. Also, both terms are in common usage (Google on either). The issue really comes down to: "is trilateration synonymous with TOA and multilateration with TDOA, or is trilateration synonymous with 3 sites and multilateration synonymous with multiple sites?". If we can resolve that, then the problem is resolved. It is clear that in normal use, "multi" is used as the general case of "tri" - eg. "trilateral agreement", "multilateral agreement". By extension, it seems illogical to describe a system with more than 3 sites using TOA as "trilateration", or (to a lesser extent) a system with exactly 3 sites as "multilateration". It also seems clear to me that even given the usual definitions of trilateration (3 sites, TOA) and multilateration (multiple sites, TDOA), the two approaches are mathematically equivalent and hence the same thing. For instance, if my classical trilateration system measures three distances, d1, d2, d3 to computer the location by the intersection of three circles, I could equally well solve the problem by taking (d2-d1) and (d2-d3) and solving for the intersection of the two resulting hyperbolae. It's simply a matter of how I choose to code up the algorithm - to the end user, nothing is different. My personal opinion is that we should explicitly address this confusion by structuring the article as I recommend below. Within the "Definition" section we could state that the most common usage of trilateration is in the context of TOA measurements, and the most common usage of multilateration is in the context of TDOA measurements. Comments? Paul 11:47, 23 January 2006 (UTC)

Re: "it seems illogical to describe a system with more than 3 sites using TOA as "trilateration". Is this really the case? Is Triangulation only triangulation when 3 angles are in use? --SC 21:00, 24 January 2006 (UTC)
Most TOA systems, deployed to date, use more then 3 sites for position fixing. In the perfect world 3 would be enough, but in the real world TOA is estimated with errors, which are caused by phenomena such as clock drifts, unknown clock biases and unknown propagation parameters. In a 2-D case with no errors in a system with 4 TOA the LOP circles would intersect in one point, and the system would be overdefined. However, in the real 2-D case with errors all 4 LOP circles normally do not intersect at one point, because errors are different for each TOA, and they would not necessarily cancel. The user would use the least-squares method (may be with a Kalman filter) to estimate her position, and the more TOAs she gets – the more accurate is her estimate. The most prominent example is the GPS, which is a TOA system, where user typically sees 5-12 satellites at the same time.
Adding to the confusion:
  • To make position fix in 3-D one needs 4 TOA in the general case.
  • One can do hyperbolic positioning with just three sites, which are arranged in three distinct pairs.
So, here’s my take on resolving this confusion:
  • Trilateration – TOA position fixing with any number of sites. We call it tri-lateration, because 3 is the minimum number of sites for the 2-D case.
  • Hyperbolic – TDOA position fixing with any number of sites.
  • Multilateration – stillborn and confusing term. No wonder that Britannica doesn’t have it. We should just warn the readers.
Kender 21:59, 24 January 2006 (UTC) Stanford, CA


To correct something above - you *can* do 2D hyperbolic positioning with 3 sites, as you you get 2 hyperboloids. However, you *can't* regard them as three distinct pairs as the TDOA of the third pair is always dependent on that from the other two. However, to describe multilateration as a "stillborn and confusing term" is utterly wrong. It may not be a term that *you* are familiar with, or used in whatever your area of expertise is, but it is in widespread use in the area of surveillance systems. Just Google and see. Hyperbolic positioning is never used in this context - even though it is describing the same concept - so I might equally well describe that as "stillborn and confusing". Whether Britannica uses it, or not, is immaterial - Wikipedia already has many more articles than Britannica and so this should not be the measure of what is included. The surveillance literature deos use the term, and that is a more important measure than Britannica (encyclopedias should never be regarded as primary sources of information). Given the obvious different usage of terms in different areas here, I suggest we stick with my suggestion below and simply state which domains most commonly use each term. Using emotive language like above is just inserting personal prejudices into a factual article. Incidentally, a large part of my day job is working with multilateration systems. They are in widespread use in civil ATC and military surveillance and are always referred to as multilateration. I'm therefore arguing this from a position of current and professional knowledge. Once again, I suggest we use my structure below and can use the "definition" and "example uses" sections to indicate which terms are in common use in each domains. Without further comment on whether we believe that domain is correct or not. Paul 06:38, 25 January 2006 (UTC)
I have also searched for "multilateration" on the IEEE site, and got 46 hits ("trilateration" 25, "hyperbolic position" 9). They seem to employ Paul's definition of multilateration. So, I take "stillborn" back. But I keep the "confusing". We should not merge the articles, and add the phrase "Mutilateration, which is based on TDOA measurements, should not be confused with trilateration, which is based on TOA measurements." to the multilateration article and remove the references to the TOA systems (such as GPS and Galileo). By the way, I work with GPS, Loran and Galileo, and, as you can see, I've never encountered the term "multilateration", so it could quite specific to the surveillance area. Kender 09:09, 25 January 2006 (UTC)
I agree with Kender - and have now changed my vote above to disagree. We should just aim to clarify both articles in light of the discussions above, and ensure the two cross-reference each other. --Paul 12:29, 25 January 2006 (UTC)

Incidentally, if we can't resolve this issue over the relation between trilateration and multilateration, I propose we abandon the proposal to merge the two articles, but simply include a "see also" at the bottom of each article. It seems that there are only three of us interested in a merge anyway... Paul 06:58, 25 January 2006 (UTC)

[edit] Proposal for Structure of Merged Articles

I propose that the two articles are merged as follows:

  • Definition (define multilateration, trilateration)
  • Principle of operation
    • Reciprocity (discussion of equivalence of multiple receivers and one transmitter, and vice versa)
    • Time of Arrival
      • Description of approach
      • Mathematics of estimation using TOA (based on existing trilateration article, but perhaps in 3D rather than 2D)
    • Time Difference of Arrival
      • Description of approach
      • Mathematics of estimation using TDOA (based on [1])
    • Phase Difference
      • Brief mention that this is equivalent to TDOA for narrowband sources
    • Multilateration accuracy (merge existing section of same name, and error model in trilateration)
  • Example applications
  • See also
  • External links

Paul 12:08, 19 January 2006 (UTC)

Propose link structures as follows:
Frelke 23:04, 19 January 2006 (UTC)

[edit] Separating Math from Measurement

I would argue that the terms trilateration and multilateration refer to the mathematical techniques used to solve for position using range or distance data - regardless of how these distances are measured. For example, what do you call the technique when it is performed with distances provided by laser range finders? With these types of measurements the concepts of TDOA and TOA are not applicable but the terms trilateration and multilateration are still used. My opinion is that multilateration is merely an extension of trilateration from the 2D case to the 3D case (as well as to situations where the localisation problem is over constrained). In my experience, these terms refer to different versions of the same technique that uses the Pythagorean theorem to form a set of non-linear equations. I think that the terms TOA and TDOA are only relevant when discussing particular positioning systems that use the time-of-flight of signals to measure distances (which includes GPS and ultrasound based systems). --Michael 17:27, 21 April 2006 (UTC)

[edit] Trilateration as described in the article can not be used for GPS purposes

This text was first put in the main article, but it was considered odd and therefore removed.

Additional, if the description of trilateration in the main article is correct, trilateration can not be used for GPS. A GPS clock (quartz clock) can be used for TDOA (Time Difference Of Arrival), but can not be used for TOA (Time Of Arrival) calculations. Trilateration as described above needs a better clock than the quartz clock available in a gps receiver. Pseudorange calculation [2][3](Pseudo-Range Navigation Solution Example), needs only a clock good enough to determine the TDOA between the different signals to calculate the position the actual time is not used. For Pseudorange calculations the signals of four satelites are needed, or the signal of tree satelites and a known hight is needed.

Comment: Only after a GPS device has made a fix and knows what it's location is, can it determine what the distances to the satellietes is. The reception of 3 satellietes allone is not enough to determine location or distances.

The quartz clock.
A clock drift of only one second a year will result in an error af about 10 meters/second so after a minute the error is as large as 600 meters. A clock drift of one second a week (more realistic) will result in 500 meters/second, this is about 100 miles per hour. This is often more than the speed we are traveling in. So if the clock can not be corrected very often, it can not be used for the positioning because the error would be larger than if the GPS device would just 'stick' to the last correct position. The clock is used for measuring the Time Difference Off Arrival (TDOA) and the accuracy of the clock is still importent, but the drift is not a problem. Crazy Software Productions 11:31, 5 September 2007 (UTC)

[edit] Why "Z" ?

Hello, I've been reading through the article because i wanted to know more about his technique. The explanatory calculus seems too complicated using 3coordinated when the article excplicitly says it is for 2D (x & y) and that z=0, would removing the Z make things clearer and more understandable ? 85.168.232.38 (talk) 03:40, 7 March 2008 (UTC)