Algorithm to find out relative location of sensors based on signal delay

Question

This relates to triangulation, but in an non intuitive way.

We have a sensor array of n sensors, that are spatially separated. They measure signal, of which speed is low enough relative to the distances between the sensors, that we can measure the delay of said signal between the sensors. (For example 4 microphones listening on sound, each spaced 1m apart or something similar).

We do not know the locations of these sensors relative to each other, nor do we know the location of the signal source, but we can move the source around the array, giving us multiple delays, that relates to the distance of the sensors relative to each other (like clapping your hands around the sensor array).

Can we find the relative positions of the array from a set of delays?

Intuitively it seems be possible, because we know the speed of signal, and delay of signal in each sensor relative to each other, so there is knowledge of the relative distance of these sensors.

One more thing: If the signal is far away, it behaves like an infinite planar source, but if the signal is close, the wavefront is spherical, and this complicates things.

And there is the 2D case and 3D case as well.

This image is of one signal measurement moment

Answer 1

Yes, with caveats.

Since you don't know the timing of the source signal you can't measure the distance directly between each array and the source.

However you can still tell the difference in distance between the different array components.

This gives you an ugly equation for each pair of array components. You can write it as something like d_1 - d_0 - measurement_delay_1_0 * signal_speed = 0, Where d_x is the distance between source and receiver (sqrt((xs-xr) ^2 + (ys-yr)^2 + (zs - zr) ^2)).

One measurement is not sufficient to solve it. So now move the source one unit in the X direction and take a new measurement. Then move the source one unit on y direction and take a measurement. And maybe one on z if you are in a 3d case. I suspect this (or 4 measurements) would be sufficient, but I didn't dug to deep into the math.

Now if this was a perfect setup, you would get perfect measurements and you should be getting a system of polynomial equations that you should be able to solve using known techniques (lookup on wikipedia or a good math book).

However, real world is not perfect so it's unlikely you would be getting an exact mathematical solution. So the next best thing is to think of it as an optimization solution where you want each equation to be optimized from a least squares perspective (again look it up). That will give you the best approximation of the locations. The nice bit is that you can increase the accuracy in this setup by taking additional measurements.

Caveats:

• Crazy detector placement might interfere with the solution. Ideally the detectors would not be co-planar.
• The coordinate system is dictated by how you move the source. Moving by very small amounts (relative) could induce large errors if the movements are very precise.
• In real world you will get echos and noise. You should deal with that before trying to locate anything.