Navigation by range measurement
In the plane, we measure the distances 
of an object located at an unknown position 
from points with known coordinates 
. The distance vector 
is a non-linear function of 
, given by
![\[\rho_i(x, y)=\sqrt{\left(x-p_i\right)^2+\left(y-q_i\right)^2}, \quad i=1, \ldots, 4 .\]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-14b538a8ed94bec33593fdccdfe63fd2_l3.png "Rendered by QuickLaTeX.com")
Now assume that we have obtained the position of the object 
at a given time and seek to predict the change in position 
that is consistent with observed small changes in the distance vector 
.
We can approximate the non-linear functions via the first-order (linear) approximation. A linearized model around a given point is 
, with 
a 
matrix with elements
![\[a_{i 1}=\frac{x_0-p_i}{\sqrt{\left(x_0-p_i\right)^2+\left(y_0-q_i\right)^2}}, \quad a_{i 2}=\frac{y_0-p_i}{\sqrt{\left(x_0-p_i\right)^2+\left(y_0-q_i\right)^2}}, \quad i=1, \ldots, 4 .\]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-d66dfd11409c4fb47ab57131a3112d93_l3.png "Rendered by QuickLaTeX.com")
