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--- cs:hydrophones:trilateration_setup:start [2016/11/15 17:11]
James Irwin [Solving for Position]
+++ cs:hydrophones:trilateration_setup:start [2017/09/12 12:44] (current)
James Irwin [Trilateration Setup]
@@ Line 1: / Line 1: @@
-====== 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.
+{{: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 =====
@@ Line 20: / Line 19: @@
 $$
-When a ping is received by the hydrophones, the hardware outputs delta-timestamps $\Delta t_x$, $\Delta t_y$, $\Delta t_z$, which corresponds to the difference in time between the ping was received by $h_0$ and $h_{x,y,z}$, respectively.
+When a ping is received by the hydrophones, the hardware outputs delta-timestamps $\Delta t_x$, $\Delta t_y$, $\Delta t_z$, which corresponds to the difference in time between when the ping was received by $h_0$ and $h_{x,y,z}$, respectively.
 Let's define $p_0$ as the absolute distance between $h_0$ and the pinger at location $(x,y,z)$.
@@ Line 33: / Line 32: @@
 \Delta z = \Delta t_z * c_s \\
 $$
-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 $\Delta z$
@@ Line 141: / Line 140: @@
 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 \\
 $$
 We can derive a similar equation for $y^2$ and $z^2$
 $$
-x^2 = p_0^2 \frac{\Delta x^2}{\delta^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 \frac{\Delta y^2}{\epsilon^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 \frac{\Delta z^2}{\zeta^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 \\
 $$
 Notice that each of these equations are in the form
 $$
-p_0^2 a + p_0 b + c
+p_0^2 e + p_0 f + g
 $$
-Lets define the following variables:
+Let's define the following variables:
 $$
-a_x = (\Delta x/\delta)^2
+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) \\
+= 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}
+= 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 =====

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