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In the plane, we measure the distances $\rho_i$ of an object located at an unknown position $(x, y)$ from points with known coordinates $\left(p_i, q_i\right), i=1, \ldots, 4$. The distance vector $\rho=\left(\rho_1, \ldots, \rho_4\right)$ is a non-linear function of $x$, given by

$$
\rho_i(x, y)=\sqrt{\left(x-p_i\right)^2+\left(y-q_i\right)^2}, \quad i=1, \ldots, 4 .
$$

Now assume that we have obtained the position of the object $\left(x_0, y_0\right)$ at a given time, and seek to predict the change in position $\delta x$ that is consistent with observed small changes in the distance vector $\delta \rho$.

We can approximate the non-linear functions $\rho_i$ via the first-order (linear) approximation. A linearized model around a given point $\left(x_0, y_0\right)$ is $\delta \rho=A \delta x$, with $A$ a $4 \times 2$ matrix with elements

$$
a_{i 1}=\frac{x_0-p_i}{\sqrt{\left(x_0-p_i\right)^2+\left(y_0-q_i\right)^2}}, \quad a_{i 2}=\frac{y_0-p_i}{\sqrt{\left(x_0-p_i\right)^2+\left(y_0-q_i\right)^2}}, \quad i=1, \ldots, 4 .
$$