I am having some trouble to find or implement an algorithm to find a signal source. The objective of my work is to find the sound emitter position.
To accomplish this I am using three vibration sensors. The technique that I am using is multilateration that is based on the time difference of arrival.
The time difference of arrival between each sensor are found using Cross Correlation of the received signals.
I already implemented the algorithm to find the time difference of arrival, but my problem is more on how multilateration works, it's unclear for me based on my reference, and I couldn't find any other good reference for this that are free/open.
I saw this post Trilateration using TDOA But I can't figure out how to solve the set of equations(7) of the wikipedia page of multilateration as i have only the three TDOA.
Any help on this would be much appreciated
You have three sensor coordinates A,B,C, unknown coordinate of signal source P, unknown time of signal start t0, and three times of signal registration ta, tb, tc.
Example: Let's sensor A caught a signal in 12:00:05, sensor B - in 12:00:00, sensor C - 12:00:07. So assign time differences: ta=5, tb=0, tc=7
Squared distances from sensors to source correspond to times of signal walk with velocity v (speed of sound in air or another environment)
(Ax-Px)^2 + (Ay-Py)^2 = (v*(ta-t0))^2 {1}
(Bx-Px)^2 + (By-Py)^2 = (v*(tb-t0))^2 {2}
(Cx-Px)^2 + (Cy-Py)^2 = (v*(tc-t0))^2 {3}
Open the brackets, then subtract equations ({2}-{1}, {3}-{2},{1}-{3}) to discard squares of unknown terms.
Ax^2-2*Ax*Px + Px^2 + Ay^2-2*Ay*Py + Py^2 = v^2*(ta^2 - 2*ta*t0 + t0^2)
Bx^2-2*Bx*Px + Px^2 + By^2-2*By*Py + Py^2 = v^2*(tb^2 - 2*tb*t0 + t0^2)
Cx^2-2*Cx*Px + Px^2 + Cy^2-2*Cy*Py + Py^2 = v^2*(tc^2 - 2*tc*t0 + t0^2)
Bx^2-Ax^2 -2*(Bx-Ax)*Px + By^2-Ay^2 -2*(By-Ay)*Py = v^2*(tb^2-ta^2 -2*(tb-ta)*t0) {1'}
Cx^2-Bx^2 -2*(Cx-Bx)*Px + Cy^2-By^2 -2*(Cy-By)*Py = v^2*(tc^2-tb^2 -2*(tc-tb)*t0) {2'}
Ax^2-Cx^2 -2*(Ax-Cx)*Px + Ay^2-Cy^2 -2*(Ay-Cy)*Py = v^2*(ta^2-tc^2 -2*(ta-tc)*t0) {3'}
Now you have system of three linear equations with three unknowns. It might be solved with some widespread algorithms - Gauss elimination, LU decomposition etc.
Note that solution precision strongly depends on small errors in coordinates and time measurements (this method is not very robust).
Geometrically, a hyperbola represents the cloud of points with a constant difference in distance between two points. You have 3 points, but taken pairwise, the time differences between the 3 possible pairs will allow you to draw 3 hyperbolas. Look for a spot at or between where the hyperbolas intersect on a plot. Or solve the equivalent algebra (least squares).
来源：https://stackoverflow.com/questions/36176167/trilateration-of-a-signal-using-time-differencetdoa