Sect_1.4.4

1.4.4 Positioning Strategies

SATELLITE-BIASED RANGE POSITIONING

The ob servation equation for a satellite-biased range is (eqn (1.3-16)):

*(t) = (t) + ^s(T_s).c

where c is the speed of electromagnetic radiation, is the satellite clock error caused by the satellite oscillator not being synchronised to "true" time, * is the measured range and is the true range. Each observation made by the receiver can be parameterised as in equation (1.4-1), except for the replacement of dt by ^s:

(x^s – x)² + (y^s – y)² + (z^s – z)² = (* – ^s.c)² (1.4-7)

Note that the time argument has been discarded.

If it is assumed that the coordinates of the satellite signal transmitter (x^s, y^s, z^s) are known, then there are six unknowns in the system: the 3-D coordinates of the receiver (x_r1, y_r1, z_r1) and the three satellite clock error terms (^si). (It is only necessary to observe to three satellites if no receiver bias is present.) Six satellite-biased range observations are therefore required in order to solve this positioning problem.

It is not feasible to simply observe more satellites, as each new satellite observation introduces a new clock parameter. It is possible, however, to take advantage of the fact that all observations made to a particular satellite are biased by the same amount (if made at the same time, or close together so that the satellite clock error can be assumed to have not changed by any appreciable amount). If three range observations are made from another station, whose coordinates are known (x_r2, y_r2, z_r2), then the following system of six equations in six unknowns is obtained:

(x^s1 - x_r1)² + (y^s1 - y_r1)² + (z^s1 - z_r1)² = (*_r1^s1 - ^s1.c)²

(x^s2 - x_r1)² + (y^s2 - y_r1)² + (z^s2 - z_r1)² = (*_r1^s2 - ^s2.c)²

(x^s3 - x_r1)² + (y^s3 - y_r1)² + (z^s3 - z_r1)² = (*_r1^s3 - ^s3.c)² (1.4-8)

(x^s1 - x_r2)² + (y^s1 - y_r2)² + (z^s1 - z_r2)² = (*_r2^s1 - ^s1.c)²

(x^s2 - x_r2)² + (y^s2 - y_r2)² + (z^s2 - z_r2)² = (*_r2^s2 - ^s2.c)²

(x^s3 - x_r2)² + (y^s3 - y_r2)² + (z^s3 - z_r2)² = (*_r2^s3 - ^s3.c)²

for which a unique solution can be obtained.

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(x^s1 - x_r1)² + (y^s1 - y_r1)² + (z^s1 - z_r1)² = (*_r1^s1 - ^s1.c)²
(x^s2 - x_r1)² + (y^s2 - y_r1)² + (z^s2 - z_r1)² = (*_r1^s2 - ^s2.c)²
(x^s3 - x_r1)² + (y^s3 - y_r1)² + (z^s3 - z_r1)² = (*_r1^s3 - ^s3.c)²	(1.4-8)
(x^s1 - x_r2)² + (y^s1 - y_r2)² + (z^s1 - z_r2)² = (*_r2^s1 - ^s1.c)²
(x^s2 - x_r2)² + (y^s2 - y_r2)² + (z^s2 - z_r2)² = (*_r2^s2 - ^s2.c)²
(x^s3 - x_r2)² + (y^s3 - y_r2)² + (z^s3 - z_r2)² = (*_r2^s3 - ^s3.c)²