Navigation by range measurement

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 (p_i,q_i) , i=1,ldots,4 . The distance vector rho = (rho_1,ldots,rho_4) is a non-linear function of , given by

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

Now assume that we have obtained the position of the object (x_0,y_0) 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 (x_0,y_0) is delta rho = A delta x , with a 4 times 2 matrix with elements

$a_{i1} = frac{x_0-p_i}{sqrt{(x_0-p_i)^2+(y_0-q_i)^2}}, ;; a_{i2} = frac{y_0-p_i}{sqrt{(x_0-p_i)^2+(y_0-q_i)^2}}, ;; i=1,ldots,4.$