Navigation by range measurement

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

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

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