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cs:hydrophones:multilateration:start [2017/09/14 00:12] Brian Moore created |
cs:hydrophones:multilateration:start [2017/09/14 00:32] Brian Moore [Multilateration] |
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| ====== Multilateration====== | ====== Multilateration====== | ||
| - | Previous derivations on this topic were made in years passed to enable calculations with limited numbers of hydrophone channels. This page provides a generalized algorithm for locating a pinger in 3D in a robust, linear manner with an arbitrary number of arbitrarily placed hydrophones. This solution is a precise calculation the evaluates at all distances, and does not require require that the signal originate in the far-field. | + | Previous derivations on this topic were made in years passed to enable calculations with limited numbers of hydrophone channels. This page provides a generalized algorithm for locating a pinger in 3D in a robust, linear manner with an arbitrary number of arbitrarily placed hydrophones. This solution is an unambiguous, precise result that evaluates at all distances and does not require that the signal originate in the far-field. |
| This method of calculation requires 5 or more hydrophones. Substituting a depth sensor for one of the required hydrophones is possible, and covered below. The submarine does //not// need to be held level to make use of the depth sensor. | This method of calculation requires 5 or more hydrophones. Substituting a depth sensor for one of the required hydrophones is possible, and covered below. The submarine does //not// need to be held level to make use of the depth sensor. | ||
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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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| \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} | \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} | ||
| \end{bmatrix}^{-1} | \end{bmatrix}^{-1} | ||
| + | \left( | ||
| \begin{bmatrix} C_2 \\ C_3 \\ z_{pinger}\\ \end{bmatrix} | \begin{bmatrix} C_2 \\ C_3 \\ z_{pinger}\\ \end{bmatrix} | ||
| - | - | ||
| - | \begin{bmatrix} B_2 \\ B_3 \\ z_{sub}\\ \end{bmatrix} \\ | + | \begin{bmatrix} B_2 \\ B_3 \\ z_{sub}\\ \end{bmatrix} |
| + | \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} \\ | ||
| $$ | $$ | ||
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| A_{zn} = det\begin{vmatrix} 2z_1 & 2z_n \\ \Delta r_1 & \Delta r_n \end{vmatrix}, \\ | A_{zn} = det\begin{vmatrix} 2z_1 & 2z_n \\ \Delta r_1 & \Delta r_n \end{vmatrix}, \\ | ||
| - | \left( | ||
| - | \begin{bmatrix} C_2 \\ C_3 \\ C_4\\ \end{bmatrix} | ||
| - | - | ||
| - | \begin{bmatrix} B_2 \\ B_3 \\ B_4\\ \end{bmatrix} | ||
| - | \right) | ||
| - | |||
| - | \left( | ||
| B_n = det\begin{vmatrix} \Delta r_1^2 & \Delta r_n^2 \\ \Delta r_1 & \Delta r_n \end{vmatrix}, \qquad | B_n = det\begin{vmatrix} \Delta r_1^2 & \Delta r_n^2 \\ \Delta r_1 & \Delta r_n \end{vmatrix}, \qquad | ||
| C_n = det\begin{vmatrix} (x_1^2+y_1^2+z_1^2) & (x_n^2+y_n^2+z_n^2) \\ \Delta r_1 & \Delta r_n \end{vmatrix} | C_n = det\begin{vmatrix} (x_1^2+y_1^2+z_1^2) & (x_n^2+y_n^2+z_n^2) \\ \Delta r_1 & \Delta r_n \end{vmatrix} | ||
| - | \right) | + | |
