GEOMETRIC PRINCIPLES |
Positioning strategies are first discussed in the context of the measurement
technology and the satellite configuration. The geometrical principles of
lines of positions and surfaces
of positions are then introduced with particular attention given
to the effect of biased ranges, as is the circumstance for GPS positioning.
A generalised algebraic description of positioning strategies appropriate
for biased ranges is then presented. In particular, the strategies are those
which account for the dominant measurement biases: receiver clock
and satellite clock errors, and measurement ambiguities
(section 6.2.1). In the case of GPS, some of these strategies are also effective
for minimising the effect of other important biases. Hence an understanding
of the concepts presented here is essential to subsequent discussions on
the operational and computational aspects of GPS, both for the navigation
and survey modes of positioning.
Measurement types :
Configurations :
POSITIONING STRATEGIES :
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Lines- and Surfaces-Of-Position
The geometrical principles of positioning can be demonstrated in terms of
the intersection of "lines-of-position" (LOP) when considered
in two-dimensions, and "surfaces-of-position" (SOP) in the case
of three-dimensional positioning.
Figure 1 below illustrates the SOPs for range measurements to a satellite -- a sphere with radius being a certain distance from the satellite; and range-difference measurements in the case of two satellites -- a hyperboloid being the locus of all points a certain distance-difference from two satellites.
Figure 1. Surfaces-Of-Position for range and range-difference measurements.
In the case of 2-D positioning the relevant geometry is defined in terms of LOPs. Position can then be defined in terms of the intersection of two sets of LOPs, involving distances to two known points, or distance-differences from three known points as shown in Figure 2. (The "known points" may be terrestrial control stations, as in the example of offshore positioning in Figure 2, or satellites.)
Figure 2. The intersection of Lines-Of-Position for 2-D positioning.
If attention is restricted to the scenario of range measurements to multiple satellites, the intersection of SOPs necessary for 3-D positioning is illustrated in Figure 3, in both the geometric and mathematical form. Note that range observations to three satellites (defining three SOPs) will solve the positioning "problem". (There are in fact two solutions to the problem, one of which will be clearly nonsense and can therefore be discarded, as in Figure 4a.)
What if the ranges are "biased" in some way? The situation
may arise as in Figure 4b! If it is assumed that the bias affects all measurements
by the same amount (for example, because it has the same physical cause),
then receiver-biased range measurements can still be used for positioning
as illustrated in Figure 5. Here the 2-D case is used to illustrate the
principle. By assuming that all measurements are biased by the same amount,
the addition of a third measurement will result in an intersection area
of a certain size. The centroid of this error figure
would therefore most likely be the correct position. This can of
course be extended to the three-dimensional case, which would require the
addition of another (biased) range measurement to a fourth satellite.
Figure 3. The geometric and mathematical problem of 3-D positioning from
ranges.
Figure 4a
Figure 4b
Intersection of SOPs: unbiased (a) and biased (b) range measurements.
Figure 5. 2-D positioning with biased ranges: the effect of adding another
measurement.
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© Chris Rizos, SNAP-UNSW, 1999
SLR