How to calculate where each sensor is when I have only few variables

Question

Suppose I have 3 sensors: sensor1, sensor2 and sensor3.

The only variables I know are:

Distance from sensor1 to origin is 36.05
Distance from sensor2 to origin is 62.00
Distance from sensor3 to origin is 63.19
Distance from sensor1 to sensor2 is 61.03
Distance from sensor1 to sensor3 is 90.07
Distance from sensor2 to sensor3 is 59.50

This is how it would look like if you had the positions:

How can I calculate the position of every point using only those variables?

This is not homework, just curiosity.

By the way, I don't even know if that's possible, I thought about this problem while taking a shower... — , Mar 20 '11 at 23:40

score 2 · Accepted Answer · answered Mar 20 '11 at 23:45

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You cannot find the position of the points exactly, as any rotation around the origin, as well as symmetry still give the same distances.

Do you want a way to find all the possible results? Finding the points is pretty straightforward, but do you need the method to be robust on noise?

answered Mar 20 '11 at 23:45

fulmicoton

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What if I assumed that the sensor1 `x` position is 0, would it be possible? – Mar 20 '11 at 23:46
You mean the origin? or some given point O? If you mean a point different from origin, you still have 2 solutions. – fulmicoton Mar 20 '11 at 23:48
I meant a point different from origin. Well, then the only way to make it only have one possibility is assuming that one point `x` position is 0.0 and its `y` position is less than zero (or greater), right? – Mar 20 '11 at 23:53
You still have the "mirror solution", don't you? – fulmicoton Mar 20 '11 at 23:55
*"You still have the "mirror solution", don't you?"* I think you might or might not – dmckee --- ex-moderator kitten Mar 21 '11 at 02:52

score 2 · Answer 2 · answered Mar 21 '11 at 09:37

This process is called trilateration. As others have noted, finding absolute, unambiguous positions for the sensors is not possible without more information - you'll need the positions of three non-coincident, non-colinear sensors in 2D, 4 non-coincident, non-coplanar sensors in 3D, to resolve all rotation/reflection ambiguities.

There's been an enormous amount of research into this problem in the field of wireless sensor network localisation - dealing with incomplete, noisy range measurements, unreliable communication and highly constrained resources make it interesting.

This might be an apt approach - the basic idea is to build up a system of located nodes piecewise - start with a seed formation of 3 or 4 nodes with well-defined relative locations and add nodes one by one as their locations become unambiguously computable relative to already-located nodes.

The anchor nodes with known locations can be used as the seed for system growth if possible, or used to compute a corrective transform after all nodes have been located.

score 1 · Answer 3 · edited May 23 '17 at 11:48

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The problem as posed is impossible without more information. If you add more information and some noise, then it is doable. See Finding a point that best fits the intersection of n spheres discusses how to solve that type of problem.

edited May 23 '17 at 11:48

Community

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answered Mar 21 '11 at 01:59

btilly

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score 0 · Answer 4 · answered Mar 25 '11 at 08:25

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Look at these images.

enter image description here

And

enter image description here

You will see that the triangle can rotate freely (so no "fixed" position exists), and also the third intersensor distance is not needed in the general case, as it is determined by the other two distances.

answered Mar 25 '11 at 08:25

Dr. belisarius

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How to calculate where each sensor is when I have only few variables

4 Answers4