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cs:hydrophones:trilateration_setup:start [2016/11/15 16:27] James Irwin [Solving for Position] |
cs:hydrophones:trilateration_setup:start [2017/09/12 12:42] James Irwin ↷ Page moved from ee:hydrophones:trilateration:start to cs:hydrophones:trilateration_setup:start |
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$$ | $$ | ||
- | When a ping is received by the hydrophones, the hardware outputs delta-timestamps $\Delta t_x$, $\Delta t_y$, $\Delta t_z$, which corresponds to the difference in time between the ping was received by $h_0$ and $h_{x,y,z}$, respectively. | + | When a ping is received by the hydrophones, the hardware outputs delta-timestamps $\Delta t_x$, $\Delta t_y$, $\Delta t_z$, which corresponds to the difference in time between when the ping was received by $h_0$ and $h_{x,y,z}$, respectively. |
Let's define $p_0$ as the absolute distance between $h_0$ and the pinger at location $(x,y,z)$. | Let's define $p_0$ as the absolute distance between $h_0$ and the pinger at location $(x,y,z)$. | ||
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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 = \frac{p_0^2 (4\Delta x^2) + p_0 (-4\Delta x^3 + 4 \delta^2 \Delta x) + (\Delta x^2 - \delta^2)^2} | x^2 = \frac{p_0^2 (4\Delta x^2) + p_0 (-4\Delta x^3 + 4 \delta^2 \Delta x) + (\Delta x^2 - \delta^2)^2} | ||
- | {4 \delta^2} | + | {4 \delta^2} \\ |
- | | + | |
+ | x^2 = p_0^2 \frac{4\Delta x^2}{4 \delta^2} + p_0 \frac{-4\Delta x^3 + 4 \delta^2 \Delta x}{4 \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 \\ | ||
$$ | $$ | ||
+ | |||
+ | We can derive a similar equation for $y^2$ and $z^2$ | ||
+ | $$ | ||
+ | x^2 = p_0^2 \left(\frac{\Delta x}{\delta}\right)^2 + p_0 \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) + \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | ||
+ | |||
+ | y^2 = p_0^2 \left(\frac{\Delta y}{\epsilon}\right)^2 + p_0 \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) + \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | ||
+ | |||
+ | z^2 = p_0^2 \left(\frac{\Delta z}{\zeta}\right)^2 + p_0 \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) + \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 \\ | ||
+ | $$ | ||
+ | |||
+ | Notice that each of these equations are in the form | ||
+ | $$ | ||
+ | p_0^2 e + p_0 f + g | ||
+ | $$ | ||
+ | |||
+ | Let's define the following variables: | ||
+ | $$ | ||
+ | a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad | ||
+ | b_x = \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) \qquad | ||
+ | c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ | ||
+ | |||
+ | a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad | ||
+ | b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) \qquad | ||
+ | c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ | ||
+ | |||
+ | a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad | ||
+ | b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) \qquad | ||
+ | c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2 | ||
+ | $$ | ||
+ | |||
+ | Therefore, we can rewrite $x^2$, $y^2$, and $z^2$ as | ||
+ | $$ \begin{equation} \label{eq:squared_pos_from_p0_simplified} | ||
+ | x^2 = p_0^2 a_x + p_0 b_x + c_x \\ | ||
+ | y^2 = p_0^2 a_y + p_0 b_y + c_y \\ | ||
+ | z^2 = p_0^2 a_z + p_0 b_z + c_z | ||
+ | \end{equation} | ||
+ | $$ | ||
+ | |||
+ | Recall that | ||
+ | $$ | ||
+ | \begin{equation} \label{eq:p_0_from_position} | ||
+ | p_0^2 = x^2 + y^z + z^2 | ||
+ | \end{equation} | ||
+ | $$ | ||
+ | Combining equations $\ref{eq:squared_pos_from_p0_simplified}$ and $\ref{eq:p_0_from_position}$, we get | ||
+ | $$ | ||
+ | p_0^2 = (p_0^2 a_x + p_0 b_x + c_x) + (p_0^2 a_y + p_0 b_y + c_y) + (p_0^2 a_z + p_0 b_z + c_z) \\ | ||
+ | p_0^2 = p_0^2(a_x + a_y + a_z) + p_0(b_x + b_y + b_z) + (c_x+c_y+c_z) \\ | ||
+ | 0 = p_0^2(a_x + a_y + a_z - 1) + p_0(b_x + b_y + b_z) + (c_x+c_y+c_z) | ||
+ | $$ | ||
+ | |||
+ | Letting | ||
+ | $$ | ||
+ | a = a_x + a_y + a_z - 1 \\ | ||
+ | b = b_x + b_y + b_z \\ | ||
+ | c = c_x + c_y + c_z | ||
+ | $$ | ||
+ | we can simplify to | ||
+ | $$ | ||
+ | \begin{equation} \label{eq:p_0_final} | ||
+ | 0 = p_0^2 a + p_0 b + c | ||
+ | \end{equation} | ||
+ | $$ | ||
+ | |||
+ | Equation $\ref{eq:p_0_final}$ is a simple quadratic equation we can use to solve for $p_0$, which will give us 2 possible values. Now that we have $p_0$, we can plug it in and solve for $x$,$y$, and $z$! | ||
+ | |||
+ | ===== Error Analysis ===== | ||
+ | |||
+ | ===== Additional Notes ===== | ||
+ | |||
+ |