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cs:hydrophones:multilateration:start [2017/09/14 00:14] Brian Moore [Multilateration] |
cs:hydrophones:multilateration:start [2017/09/14 00:29] Brian Moore [Rotation] |
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| In the event the submarine is rotated off of the default orientation $R(\psi,\phi,\theta) = R(0,0,0)$ then the relative estimation of the pinger's position will be off, and a correction must be made to get the absolute coordinates $p_{pinger}=(X,Y,Z)$. | In the event the submarine is rotated off of the default orientation $R(\psi,\phi,\theta) = R(0,0,0)$ then the relative estimation of the pinger's position will be off, and a correction must be made to get the absolute coordinates $p_{pinger}=(X,Y,Z)$. | ||
| - | If the array is rotated by $R$ then the relative position of the pinger has been rotated in the local frame by $R^T$ | + | If the array is rotated by $R$ then the relative position of the pinger has been rotated in the local frame by the inverse $R^T$ |
| $$ | $$ | ||
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| $$ | $$ | ||
| - | Which requires only 4 hydrophones, $h_0,h_1,h_2,$ and $h_3$ plus the depth sensor measuring $z_{sub}$ to compared against the known depth of the pinger $z_pinger$. | + | Which requires only 4 hydrophones, $h_0,h_1,h_2,$ and $h_3$ plus the depth sensor measuring $z_{sub}$ to compared against the known depth of the pinger $z_{pinger}$. |
| However, the depth sensor will always measure the absolute depth of the submarine regardless of the submarine's orientation $R(\psi,\phi,\theta)$. Thus the difference measured will always be $z_{pinger}-z_{sub} = Z_{global}$. | However, the depth sensor will always measure the absolute depth of the submarine regardless of the submarine's orientation $R(\psi,\phi,\theta)$. Thus the difference measured will always be $z_{pinger}-z_{sub} = Z_{global}$. | ||
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| \right) | \right) | ||
| \\ | \\ | ||
| + | $$ | ||
| + | |||
| + | With polar bearings: | ||
| + | |||
| + | $$ | ||
| + | Azimuth = arctan\left(\frac{Y}{X}\right), \qquad | ||
| + | Inclination = arctan\left(\frac{Z}{\sqrt{X^2+Y^2}}\right), \qquad | ||
| + | Range = \sqrt{X^2 + Y^2 + Z^2} \\ | ||
| + | |||
| $$ | $$ | ||
