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please help to understand the concept behind 3d multilateration (screenshot attached)

please find attached the tutorial if the screenshot does not make sense

ok... this is from Matlab and shows the intersection of the hyperbolas at the point (4,8) to show the location of TX while f1, f2, f3, f4 are RX's.

This gives a location ok for x,y or x,y,0

but what about if I wish to locate a point at x,y,z.

this is done with hyperboloids intersecting?

Can someone please explain what would be required and the concept behind it?

thanks in advance.


screenshot.jpg
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jtiernan2008
Asked:
jtiernan2008
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jtiernan2008Author Commented:
tutorial attached below;
toa.pdf
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aburrCommented:
"this is done with hyperboloids intersecting?"
yes
The concepts are the same as in the 2D case. There is a relation between distance and time. The details are in the tutorial. I know, it takes som,e study and practice. But the tutorial is 12 pages long. One cannot expand on everything in it without typing 24 pages.
Now if you have a specific question perhaps it can be answered in fewer pages.
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jtiernan2008Author Commented:
I dont see anything about it in the tutorial? I just refers to it at the start.

"The manner in which the arrival time mea-
surements locate the source can be determined
graphically from the fact that the differences inhe arrival times at a pair of stations i,j con-
strain the source to lie on a hyperboloid of rev-
olution about the baseline between the two sta-
tions."

"to lie on a hyperboloid of revolution" - what does this mean?

Do you know any reading material on this like the pdf that shows how to plot and find the intersection of these hyperboloids?
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jtiernan2008Author Commented:
I understand the 2D version (x,y) fine as explained in the tutorial not the 3D version (x,y,z)
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yuk99Commented:
Hyperboloid with baseline along z can be described by equation:
z^2/a^2 - x^2/b^2-y^2/c^2=1
Hyperboloid of rotation is hyperboloid with b=c. So at z>a its intersection with xy plane will be a circle.
Remember when you derived hyperbola playing with circles of different radii with the same difference? With 3D you have one more degree of freedom. So you have to do the same, but now with spheres with centers located on z axis.
Also note that rotation of hyperboloid in 3D is more complex then what you did in 2D.
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jtiernan2008Author Commented:
Thanks for the great explanation... I appreciate it and understand what you are saying...
Do you know any reading material on this?
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yuk99Commented:
For TOA in 3D I don't know.
For mathematical concepts of hyperboloid, just google it. You will find a lot of material, including wikipedia, mathworld, etc.
Rotation in 3D and rotation matrices are also easy to find there.
Good luck!
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