My two last days were dedicated to looking for a proper triangulation algorithm.
Also the web ives many results for “triangulation” or “localisation” combined with “ultrasonics”, “equations”, “time of flight” and many other key words, I didn’t find the equations we were looking for. Lots of article deals with recovering a position from angle measure but not from time of flights or distance measure.
The best solution I found, is writing the equation myself using Maple software to solve them.
Good point is that Maple is able to solve those equations, bad news is that the result for each coordinate is about 5 pages long and has an imaginary part.
I hope that I will be able to decipher the result and implement the algorithm for a simulation soon.