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•  » Calculating 2D position from travel time without knowing start time

#1 2013-11-02 06:19:52

KDM
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Calculating 2D position from travel time without knowing start time

Hi.

First post. Hope it's worthy.

I need to calculate the position in 2D space of an "event" using four sensors positioned at the corners of the rectangular "universe".

Let's say the event is a sound which might occur somewhere inside the rectangle and the sensors are microphones.

Is it possible to calculate the co-ordinates for the epicentre based on the time for the sound to reach each of the four sensors?

At first sight, it seems straightforward enough. Until you realise that you don't know what time the event happened at, so you don't actually know the time for the event to travel to any of the sensors. You only know the time delta between the sound hitting each of the sensors.

So, the event happens at time S (but we don't know when that was). Then (in the shown example) the soundwave hits B A C and D. We can see that the timestamp registered at point C is going to be the time SC minus the time SB, etc:

tC = SC - SB and
tD = SD - SB
tA = SA - SB

But we don't know time SB.

Can anyone help me find the location of S, please?

Rules suggest I should indicate my "level". Well, I didn't sit high school maths, but I'm a masters degree qualified Aerospace engineer. So, I'm a bit irked that I couldn't work this one out for myself...

Last edited by KDM (2013-11-02 06:27:41)

#2 2013-11-02 06:33:49

bob bundy
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Re: Calculating 2D position from travel time without knowing start time

hi KDM,

Welcome to the forum!

Of course it is worthy.  But that doesn't mean we can do it.  I'm still working on a problem set over two years ago.  I'll start thinking about it.  Meanwhile, someone else will probably nip in and provide a solution.

Back later.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#3 2013-11-02 07:14:00

KDM
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Re: Calculating 2D position from travel time without knowing start time

Thanks. I know it can be done. I'm pretty sure I've seen ultrasound used to triangulate swarm robots without knowing an angle: just time deltas. I just can't find it again!! But, don't you hate when someone else swoops in with an answer? Heh heh. I feel your pain!

#4 2013-11-02 08:14:03

bob bundy
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Re: Calculating 2D position from travel time without knowing start time

Hi KDM,

I've hit an algebraic barrier, and, so far, cannot get round it.

Let sound be at time T, and the times at the points be Ta, Tb etc.  Speed of sound, s.  Then

From which T can be eliminated thus:

Trouble is there are 5 unknowns and only 4 equations.

I'm wondering if there is a way to compute these by trial and improvement.

Anyway, I haven't stopped thinking ........................

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#5 2013-11-02 08:25:13

KDM
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Re: Calculating 2D position from travel time without knowing start time

I have some calculated data. One thing I can do, based on knowing the co-ordinates of the event is calculate the delta time (assuming speed = 1) between the first sensor and the other three. Now, how do I attach a spreadsheet?

Code:

```x  y     tB     tC      tD      tA
4   4  10.84  16.97   10.84    0.00
4   8   3.70  11.06    8.94    0.00
8   4   8.94  11.06    3.70    0.00
8   8   3.11   5.66    3.11    0.00
3  12   0.00  10.24   12.26    3.83
3  16   0.00  12.46   18.35   11.28
6  18   0.00   7.82   16.48   12.65
9  12   0.00   1.56    4.24    2.96
9  16   0.00   1.86    9.57    8.51
10  10   0.00   0.00    0.00    0.00
11   2  11.88  10.91    0.00    1.96
14   5  12.71   8.35    0.00    7.06
16   3  18.35  12.46    0.00   11.28
17   2  21.15  14.64    0.00   13.51
18   6  16.48   7.82    0.00   12.65
14  14   6.75   0.00    6.75   11.31
14  18   7.82   0.00   12.65   16.48
16  16  10.84   0.00   10.84   16.97
18  14  12.65   0.00    7.82   16.48
18  18  15.28   0.00   15.28   22.63```

Oh, just to help, I changed it to a 20x20 universe instead of 10x10. :-(

Last edited by KDM (2013-11-02 08:30:01)

#6 2013-11-02 08:40:34

bob bundy
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Re: Calculating 2D position from travel time without knowing start time

I resorted to google and found this

http://math.stackexchange.com/questions … angulation

But you need a maths package to solve the equations.

I don't think there is a way to post a spreadsheet.  I take screen shots and post the resulting images.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#7 2013-11-02 09:23:53

KDM
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Re: Calculating 2D position from travel time without knowing start time

That does look like exactly the same problem. OKay, the solution assumes the sensors are colinear. That;s not a big issue. Will read this more thoroughly after a few hours Zzz..

#8 2013-11-02 12:58:55

bobbym

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Re: Calculating 2D position from travel time without knowing start time

Hi;

I have solved a problem like this using two methods. One is an iterative approach and the other requires solving a 3 x 3 set of equations all of which are circles.

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#9 2013-11-02 22:10:49

KDM
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Re: Calculating 2D position from travel time without knowing start time

Do you think you could do it again, please?

#10 2013-11-03 00:12:31

bobbym

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Re: Calculating 2D position from travel time without knowing start time

Hi;

This can be solved easily and is very similar to your problem:

Q: Four listening posts are stationed in the Pacific 1000 miles apart at A,B,C and D. They form a 1000 mile square. A nuclear explosion of unknown origin is detonated inside the square. The sound detectors are very sensitive and pickup the sounds according to this data

C gets the sound at 3:00 am exactly.
D gets the sound 28.707489170445314 minutes later.
B gets the sound 38.581659920734374 minutes later.
A gets the sound 54.87964482242969 minutes later.

The speed of sound was given as 768 mph.

Where in the square does the explosion occur and when?

The equations become:

x1 = 700

y1 = 800

distc = 360.55512754634225 miles

It is now easy to answer the remaining question.

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#11 2013-11-05 06:12:26

KDM
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Re: Calculating 2D position from travel time without knowing start time

Ah, fantastic. Looks like exactly what I need. And a frighteningly similar example of why ...!
Let me work that through a few times.
Thanks.

#12 2013-11-05 08:25:46

bobbym

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Re: Calculating 2D position from travel time without knowing start time

Hi;

Okay, let me know what you need to solve it.

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#13 2013-11-16 23:54:00

KDM
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Re: Calculating 2D position from travel time without knowing start time

I need to go back 25 years to school and pay more attention to simultaneous equations.
I'm going to remove speed (speed = 1) and make the distance unitless.

Last edited by KDM (2013-11-16 23:55:19)

#14 2013-11-16 23:55:40

bobbym

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Re: Calculating 2D position from travel time without knowing start time

So does everyone else, but that does not stop anyone from doing math. So do not worry, what can I help with?

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
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