2D trilateration with 3 sensors

[Ischyros]

Hello science folks!

I've just started learning about trilateration. I am wondering if its possible to calculate the coordinates of a continuously growing circle(for example a sound wave) which hits three sensors at different times. I have added a picture so you can see what I mean.
In the Wikipedia article they have both the coordinates for each sensor AND some kind of radius for each sensor. In my case I will have the coordinates for each sensor but not the radius. Instead I will know the time it takes for the wave to reach sensor#2 & #3, the time starts when the wave hits sensor #1.(look at the picture)
The speed of the wave is also unknown but to get some theoretical measurements I have set it to √8 m/s, so it will take exactly 1 second for the wave to travel to sensor #1.

So, known data:

• Coordinates of the three sensors
• time, started when sensor #1 detects the wave

Unknown data:

• Velocity of the wave
• origin of the wave

I hope you understand my question and I appreciate all the help I can get! Take an extra look at the picture if you don't understand.

 

[Simon Bridge]

Welcome to PF;

I am wondering if its possible to calculate the coordinates of a continuously growing circle(for example a sound wave) which hits three sensors at different times.

Yes it is possible.

Best set up the problem like this - the pulse, starting at time T0, is a circle with origin (x0,y0) whose radius is expanding at a constant speed v. Sensors 1,2,3 are at positions (x1,y1),(x2,y2) and (x3,y3). They receive the pulse at times T1, T2, T3, respectively.

You need to find (x0,y0) (and, presumably, T0 and v) from knowing the rest.

It can help you understand what is needed by imagining you are lucky enough for two of your sensors to be in line with the source of the pulse. Would your task be easier or harder if all three are in line with the source?

 

[Ischyros]

First of all thank you, this forum really seems like the place I've been looking for.

Best set up the problem like this - the pulse, starting at time T0, is a circle with origin (x0,y0) whose radius is expanding at a constant speed v. Sensors 1,2,3 are at positions (x1,y1),(x2,y2) and (x3,y3). They receive the pulse at times T1, T2, T3, respectively.

Allright, I will change my setup so the origin of the pulse is the origin of the coordinate system. This is also a more likely practical scenario for me than having the origin of the pulse "within" a triangle of the three sensors. I made that setup because then I would be able to apply the formulas and calculations from the wikipedia article but anyway this seems better.
But I just realized one thing, if the source of the pulse is at the oirgin of the coordinate system then the coordinates for the sensors (x1,y1) etc. is unknown. But I will know the distance between the sensors cause I arranged them, right?

You need to find (x0,y0) (and, presumably, T0 and v) from knowing the rest.

It can help you understand what is needed by imagining you are lucky enough for two of your sensors to be in line with the source of the pulse. Would your task be easier or harder if all three are in line with the source?

If two or three sensors are in line with the source I can only figger out a way to calculate velocity v and not (x0,y0) or T0. The velocity itself won't help me much cause the source of the wave could be anywhere and still have the same velocity.
I will change my picture posted in the first post but how do I proceed? the difficult thing to handle in this problem seem to be the time that has gone before the pulse hits sensor #1. The radius when the time is T1 is also a problem. thanks again!

 

[Simon Bridge]

Your general setup will have the origin of the pulse pretty much anywhere - not at the origin of your coordinate system.

These setups may be used to locate a shooter from a gunshot against a city grid - it seems a but optimistic that the (0,0) for the detector grid happens to be where the shooter is. (It's also how to do passive triangulation with cell phones.)

However, starting out with a simpler geometry will help you understand the problem better.

Note: you normally know the speed of the pulse. You are not so much interested in locating the exact few cubic centimeters the sound came from as giving a large body of searchers a good idea where to look.

To proceed you need to use the difference in times between sensors and your knowledge of how the sensors are distributed to work out the directions the pulse could have come from.

In the case where two sensors happen to be in line with the origin of the pulse, you can easily work out the speed of the pulse and the direction it came from. i.e. if S1 and S2 are in line with the center, and T1<T2, then S1 is closer. The origin must be on the line joining S2 to S1 and in that direction.

If T3=T1 then you know that S3 is exactly the same distance away from the center as S1. If T3 > T1 then you know S3 is further away, and the radial separation between S3 and S1 is going to be v(T3-T1). So find the point on the line that gives you that radial separation.

 

[dodo]

I was thinking (without having worked anything out, just idle brainstorming) that another way of setting up the problem is to start a circular wave at time 0, centered at the first sensor and expanding. Then, after a certain (known) time, another circular wave centered at sensor 2 (or 3) starts and expands; then, after some other (known) time, a third wave expands from the remaining sensor. The problem would be to find the time t that minimizes the distance between intersections of the circles.

The speed should not have to be an unknown - knowing the nature of the wave (sound, EM) and the medium (air, water) you should be able to propose a value.

P.S.: Did a small computer simulation to illustrate the point.

With sensors at positions (0,0), (10,0) and (10,10), and wave speed = 1 unit/second,
and assuming that the times when sensors 2 and 3 are triggered (after sensor 1 is) are resp. 2 and 3 seconds,
the following graph illustrates the minimal distance between circle intersections, as a function of the time to trigger sensor 1:

giving a solution at an approx. position of (3.68, 4.19).

 

[Ischyros]

Your general setup will have the origin of the pulse pretty much anywhere - not at the origin of your coordinate system.

I understood that just after I created the picture. But let's say for this time the place of the origin will be the coordinate systems origin so if we will manage to calculate its coordinates we will get (0,0).

These setups may be used to locate a shooter from a gunshot against a city grid - it seems a but optimistic that the (0,0) for the detector grid happens to be where the shooter is. (It's also how to do passive triangulation with cell phones.)

However, starting out with a simpler geometry will help you understand the problem better.

Note: you normally know the speed of the pulse. You are not so much interested in locating the exact few cubic centimeters the sound came from as giving a large body of searchers a good idea where to look.

Yes it could be a sound wave and that would be very comfortable cause we know that the speed of sound is around 340 m/s. BUT I imagine my three sensors being able to detect vibration and not sound. So the origin of the pulse could be a knock on the surface.

To proceed you need to use the difference in times between sensors and your knowledge of how the sensors are distributed to work out the directions the pulse could have come from.

In the case where two sensors happen to be in line with the origin of the pulse, you can easily work out the speed of the pulse and the direction it came from. i.e. if S1 and S2 are in line with the center, and T1<T2, then S1 is closer. The origin must be on the line joining S2 to S1 and in that direction.

If T3=T1 then you know that S3 is exactly the same distance away from the center as S1. If T3 > T1 then you know S3 is further away, and the radial separation between S3 and S1 is going to be v(T3-T1). So find the point on the line that gives you that radial separation.

Okay so first of all i will setup some kind of rules to handle which direction the pulse come from? if it hits S2 first its closer to S2 and so on as you explain. But what do you mean by radial separation? Like in my example T1 = 0s. because that's the starting time when it hits S1.
T3 = 2.66s. and based on the data from S1 and S2 the speed of the pulse is 1 unit/second.
so your radial separation would be 1*(2.66-0)=2.66 units?