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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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cs:hydrophones:trilateration_setup:start [2016/11/16 16:45] Ryan Summers |
cs:hydrophones:trilateration_setup:start [2017/09/12 12:44] (current) James Irwin [Trilateration Setup] |
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| - | ====== Trilateration ====== | + | ====== Trilateration Setup====== |
| Below is the math for calculating the location of the pinger in the water relative to our submarine. Because we know the absolute location of the pinger in the pool, we can calculate the sub's position in the pool. | Below is the math for calculating the location of the pinger in the water relative to our submarine. Because we know the absolute location of the pinger in the pool, we can calculate the sub's position in the pool. | ||
| - | + | {{:cs:hydrophones:trilateration_setup:trilateration_derivation.pdf|Original derivation}} by Brian Moore. | |
| - | {{:ee:hydrophones:trilateration:trilateration_derivation.pdf | Original derivation}} by Brian Moore. | + | |
| ===== Problem Setup ===== | ===== Problem Setup ===== | ||
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| \Delta z = \Delta t_z * c_s \\ | \Delta z = \Delta t_z * c_s \\ | ||
| $$ | $$ | ||
| - | In other words, $h_x$ is $\Delta x$ meters farther from the pinger than $h_0$, and $h_0$ is $p_0$ meters from the pinger. | + | In other words, $h_x$ is $\Delta x$ meters closer to the pinger than $h_0$, and $h_0$ is $p_0$ meters from the pinger. |
| The final calculations for $x$, $y$, and $z$ will be in terms of $\Delta x$, $\Delta y$, and $\Delta z$ | The final calculations for $x$, $y$, and $z$ will be in terms of $\Delta x$, $\Delta y$, and $\Delta z$ | ||
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| x^2 = p_0^2 \frac{4\Delta x^2}{4 \delta^2} + p_0 \frac{4\Delta x}{4\delta^2}(-\Delta x^2 + \delta^2) + \frac{(\Delta x^2 - \delta^2)^2}{4 \delta^2} \\ | x^2 = p_0^2 \frac{4\Delta x^2}{4 \delta^2} + p_0 \frac{4\Delta x}{4\delta^2}(-\Delta x^2 + \delta^2) + \frac{(\Delta x^2 - \delta^2)^2}{4 \delta^2} \\ | ||
| - | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 +\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | + | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ |
| $$ | $$ | ||
| We can derive a similar equation for $y^2$ and $z^2$ | We can derive a similar equation for $y^2$ and $z^2$ | ||
| $$ | $$ | ||
| - | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 +\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | + | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ |
| - | y^2 = p_0^2 \left(\frac{\Delta y}{\epsilon}\right)^2 + p_0 \frac{\Delta y}{\epsilon^2}(\epsilon^2 +\Delta y^2) + \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | + | y^2 = p_0^2 \left(\frac{\Delta y}{\epsilon}\right)^2 + p_0 \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) + \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ |
| - | z^2 = p_0^2 \left(\frac{\Delta z}{\zeta}\right)^2 + p_0 \frac{\Delta z}{\zeta^2}(\zeta^2 +\Delta z^2) + \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 \\ | + | z^2 = p_0^2 \left(\frac{\Delta z}{\zeta}\right)^2 + p_0 \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) + \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 \\ |
| $$ | $$ | ||
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| $$ | $$ | ||
| a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad | a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad | ||
| - | b_x = \frac{\Delta x}{\delta^2}(\delta^2 +\Delta x^2) \qquad | + | b_x = \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) \qquad |
| c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | ||
| a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad | a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad | ||
| - | b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 +\Delta y^2) \qquad | + | b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) \qquad |
| c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | ||
| a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad | a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad | ||
| - | b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 +\Delta z^2) \qquad | + | b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) \qquad |
| c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 | c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 | ||
| $$ | $$ | ||
