Linearizing GPS equations In order to find position using the GPS system, we need to know the location of at least three satellites and the distance to each of those satellites. Assume that the three satellites are located respectively at (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), and that the distances between us and the three satellites are respectively d1, d2, and d3. The following non-linear system of equations needs to be solved,

ðx
x1Þ2 þ ðy
y1Þ2 þ ðz
z1Þ2 ¼ d 21

;

ðx
x2Þ2 þ ðy
y2Þ2 þ ðz
z2Þ2 ¼ d 22

;

ðx
x3Þ2 þ ðy
y3Þ2 þ ðz
z3Þ2 ¼ d 23

:

Obviously linearization is desirable in this case. Assume that the reference point is (0,0,0). Prove that the resulting system after linearizing (9.40) is x1 y1 z1 x2 y2 z2 x3 y3 y3 2 4 3 5 x y z 2 4 3 5 ¼
x21 þ y21 þ z21
d2 1 x22 þ y22 þ z22
d2 2 x23 þ y23 þ z23
d2 3 2 4 3 5: