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November 8[edit]

Having problem implementing equation 7 in Multilateration[edit]

Question moved from WP:RD/C Tevildo (talk) 19:52, 8 November 2017 (UTC)[reply]

I have been having great difficultly implementing equation 7 in the article.

I have 3 receivers. I have modified equation 7 by simply removing the z terms (and Cm), as I can only do a 2D fix with 3 receivers. So I have receivers with subscript 0, 1 and 2 as per Fig 2.

When you substitute the values into equation 7, I only have a single set of data m=2 to substitute. This gives me 1 equation with 2 unknowns (x,y). There will be an infinite number of solutions. I must be missing something as vt1=v(T1-T0) and vt2=v(T2-T0), so I only have 2 vt values to work with, but I can't see how to get 2 equations out of it.

Help. — Preceding unsigned comment added by Mhillman (talk • contribs) 16:02, 8 November 2017 (UTC)[reply]

It seems that there is more going on here than the article (which seems poorly referenced imo) is letting on. For 3 receivers it appears you do get a single linear equation when you eliminate the R₀ terms, and this narrows the possible solutions to a line. You still have the original hyperbola equations though, pick one of these at find the two points of intersection of it and the line. This gives two possible solutions but presumably only one will be on the correct branch of the hyperbola in question. (The difference in times narrows the position of the emitter not just to a hyperbola, but to a specific branch of a hyperbola.) For 4 receivers you would get 2 linear equation in 2 unknowns, which gives a unique solution, but the original system is overdetermined so the question arises how to find the most likely estimate from the given data. For 5 or more receivers even the linear system is overdetermined. I can only speak to the analytic geometry issue so I'd suggest finding an actual textbook or journal article which covers the topic. --RDBury (talk) 22:31, 8 November 2017 (UTC)[reply]