Hello,
this is my first post here, I read the rules for new posters, non-native speaker, so I'm not completely sure whether I'm not doing something completely wrong
Background:
I'm solving practical problem of navigating a robot using time-of-flight of ultrasound waves. It is kind of similar to GPS, but much more crude and using slower propagating waves, allowing much simpler electronics, perhaps there are other advantages to be discovered. Using intuition I came to conclusion that I need three ultrasound transmitters for one receiver. The transmitters are perfectly synchronized to each other, but receiver has no precise timebase, though it can measure distance between ultrasound beeps with reasonable accuracy. I'm good at electronics and practical side of things, but I realized how badly I failed with math.
Plan goes like this: Three transmitters A, B and C are located in known positions. Transmitter A makes a short beep, silences itself and waits. After a known time, transmitter B makes a beep, waits, then C makes a beep and cycle can repeat. Receiver R doesn't know the exact time of when A beeped, but can measure time difference between A and expected beep B, as well as difference between A and C. I assume those two time delays should be enough to calculate two dimensional position of receiver R relative to A, B and C.
Reduction to trigonometry and later algebraic expressions:
For sake of simplicity, lets forget there are time differences between beeps and just assume all three transmitters do a single beep at the same time and receiver can somehow distinguish between them. I drew a crude picture, please see the attachment. If the picture is unreadable, let me know, I can redraw it. It also covers the part of calculation I can do, but got stuck later.
There are Cartesian coordinates with four points: A, B, C and R. A is always at (0,0), B is always at (By,0) and c is always at (0,Cx). By and Cx are known constants. R unknown and I need to calculate its position (coordinates Rx and Ry).
Equations 1A, 1B and 1C are my take on calculating the lengths from receiver to point A, B and C respectively.
Sound wave from A takes time T0 to arrive to receiver R. T0 is unknown to receiver. Sound wave from B takes (T0+D1) that is, it hits receiver D1 difference later than T0 (ie sound from A). Similarly, sound wave from C hits receiver at time T0+D2. Equations 2A-2C capture this, plus it is multiplied by S (speed of sound, it is known constant), so that lengths AR, BR and CR are expressed as time of flight of respective sound wave.
From equations 1 and 2 I combined equations 3 to get set of three equations with three unknown variables - I thought this should be simple now.
I entered the equation 3A into equations 3B and 3C and got 4A and 4B. Since T0 is unknown, I expressed it from 3A as equation 5 and made substitution in 4A and 4B, got results 6A and 6B. Both equations looks somehow reasonable and "symmetric", but I don't know how to proceed from this point on.
Expected result:
My ultimate goal is to calculate Rx and Ry, from known values S, By, Cx, D1 and D2. T0 is unknown, but I hoped it will dissapear at the end (it should be irrelevant since I have three transmitters). I'd say that equations 1 and 2 are OK, but perhaps the rest could be done differently.
Thank you for your help.
this is my first post here, I read the rules for new posters, non-native speaker, so I'm not completely sure whether I'm not doing something completely wrong
Background:
I'm solving practical problem of navigating a robot using time-of-flight of ultrasound waves. It is kind of similar to GPS, but much more crude and using slower propagating waves, allowing much simpler electronics, perhaps there are other advantages to be discovered. Using intuition I came to conclusion that I need three ultrasound transmitters for one receiver. The transmitters are perfectly synchronized to each other, but receiver has no precise timebase, though it can measure distance between ultrasound beeps with reasonable accuracy. I'm good at electronics and practical side of things, but I realized how badly I failed with math.
Plan goes like this: Three transmitters A, B and C are located in known positions. Transmitter A makes a short beep, silences itself and waits. After a known time, transmitter B makes a beep, waits, then C makes a beep and cycle can repeat. Receiver R doesn't know the exact time of when A beeped, but can measure time difference between A and expected beep B, as well as difference between A and C. I assume those two time delays should be enough to calculate two dimensional position of receiver R relative to A, B and C.
Reduction to trigonometry and later algebraic expressions:
For sake of simplicity, lets forget there are time differences between beeps and just assume all three transmitters do a single beep at the same time and receiver can somehow distinguish between them. I drew a crude picture, please see the attachment. If the picture is unreadable, let me know, I can redraw it. It also covers the part of calculation I can do, but got stuck later.
There are Cartesian coordinates with four points: A, B, C and R. A is always at (0,0), B is always at (By,0) and c is always at (0,Cx). By and Cx are known constants. R unknown and I need to calculate its position (coordinates Rx and Ry).
Equations 1A, 1B and 1C are my take on calculating the lengths from receiver to point A, B and C respectively.
Sound wave from A takes time T0 to arrive to receiver R. T0 is unknown to receiver. Sound wave from B takes (T0+D1) that is, it hits receiver D1 difference later than T0 (ie sound from A). Similarly, sound wave from C hits receiver at time T0+D2. Equations 2A-2C capture this, plus it is multiplied by S (speed of sound, it is known constant), so that lengths AR, BR and CR are expressed as time of flight of respective sound wave.
From equations 1 and 2 I combined equations 3 to get set of three equations with three unknown variables - I thought this should be simple now.
I entered the equation 3A into equations 3B and 3C and got 4A and 4B. Since T0 is unknown, I expressed it from 3A as equation 5 and made substitution in 4A and 4B, got results 6A and 6B. Both equations looks somehow reasonable and "symmetric", but I don't know how to proceed from this point on.
Expected result:
My ultimate goal is to calculate Rx and Ry, from known values S, By, Cx, D1 and D2. T0 is unknown, but I hoped it will dissapear at the end (it should be irrelevant since I have three transmitters). I'd say that equations 1 and 2 are OK, but perhaps the rest could be done differently.
Thank you for your help.