RANGE POSITIONING WITH
|
The observation equation for an ambiguous range, where the bias is a constant, is (eqn (1.3-19)):
 *(t)
= (t) - nr1si |
where n is the bias that is specific to a receiver (rj) - satellite (si)
pair, * is the measured range and is the true range. Note that there
is no time dependence for n. Each observation made by the receiver can be
parameterised by a modification of eqn (1.4-1):
| (xs(t) – x)2 + (ys(t) – y)2
+ (zs(t) – z)2 = (*(t) – nsi)2 |
(1.4-10) |
If it is assumed that the coordinates of the satellite signal transmitter (xs(t), ys(t), zs(t)) are known, then there are six unknowns in the system: the 3-D coordinates of the receiver (x, y, z) and the three bias terms (nsi). (It is only necessary to observe to three satellites if no receiver bias is present.) Therefore six ambiguous range observations are required in order to solve this positioning problem. It is not possible to simply observe more satellites, as each new satellite observation introduces a new ambiguity parameter. It is possible, however, to take advantage of the fact that all observations made from a certain receiver to a particular satellite are biased by the same amount. If three more range observations are made from the receiver to the same three satellites, but at some time interval later, the system of six equations in six unknowns can be obtained:
| (xs1(t) – x)2 + (ys1(t) – y)2
+ (zs1(t) – z)2 = (*s1(t) –
ns1)2 |
|
| (xs2(t) – x)2 + (ys2(t) – y)2
+ (zs2(t) – z)2 = (*s2(t) –
ns2)2 |
|
| (xs3(t) – x)2 + (ys3(t) – y)2
+ (zs3(t) – z)2 = (*s3(t) –
ns3)2 |
(1.4-11) |
| (xs1(t + dt) – x)2 + (ys1(t + dt)
– y)2 + (zs1(t + dt) – z)2 =
(*s1(t
+ dt) – ns1)2 |
|
| (xs2(t + dt) – x)2 + (ys2(t + dt)
– y)2 + (zs2(t + dt) – z)2 =
(*s2(t
+ dt) – ns2)2 |
|
| (xs3(t + dt) – x)2 + (ys3(t + dt)
– y)2 + (zs3(t + dt) – z)2 =
(*s3(t
+ dt) – ns3)2 |
for which a unique solution can be obtained. Note, it is assumed
that the receiver has not moved between time t and t+dt.
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© Chris Rizos, SNAP-UNSW, 1999