POSITIONING STRATEGIES |
In considering appropriate positioning strategies for the measurements described
above it is necessary to make some assumptions, primarily:
That the satellite coordinates are available.
The basic concept of positioning using range data is the same whether
it is involves terrestrial distances or satellite measurements. In
three dimensions, a measured range to a known point constrains the position
in 3-D space to lie on the surface of a sphere centred at the known point.
This is the "surface of position". The intersection
of three such surfaces describes a unique point in space (Figure 1). Hence,
three ranges are required, to three separated known points, in order to
fix position (section 1.4.1).
In 2-D positioning applications, for example as in the case of horizontal geodetic networks or when navigating at sea (position is assumed to be on the surface of the ellipsoid, or a known height above it), the intersection of the surface of position and the ellipsoid is the "line of position", and is approximately a circle (section 1.4.1). The intersection of two distances defines the location of the point from which the distance measurements were made. In surveying parlance, this is known simply as distance intersection.
If the point is stationary, the two (or three) distances do not need to be measured simultaneously. If the point is moving however, all distances must be measured simultaneously, or over an interval of time during which the point has not moved by an amount greater than the uncertainty of the "fix". Satellite positioning using ranges is the basis of the GPS system for most navigation applications.
Figure 1. Intersection of "surfaces of position" based on range
observations.
As with ranging, the concept of positioning using range-difference (or "range-rate")
data is the same whether based on terrestrial or satellite measurements.
Differences in range measured to two known points constrains the position
in 3-D to lie on a surface of position which is one half of a hyperboloid
of revolution of two sheets. In 2-D the line of position (formed by the
intersection of the hyperboloid and the ellipsoid) is approximately one
half of a hyperbola (see Figure 2 below).
Figure 2. Hyperbolae and hyperbolic intersection.
In the 3-D case, the intersection of three hyperboloids defines the position of the observer. Each range-difference involves two known points, and since one of the points can be common to other pairs of stations, a minimum of four known points are required for positioning based on range-rate. In the 2-D case, the intersection of two hyperbolas is sufficient for position determination, involving a minimum of three known points. This is the basis of radio-navigation systems such as LORAN-C, operating in the range-difference mode.
The TRANSIT Doppler system operates on a similar principle (see, for example, SEEBER, 1993, for a good overview), except that the two known points are in fact generated by the one moving satellite. The range-difference therefore involves the same satellite.
As in the case of positioning by range, if the point being positioned is not moving, the measurements need not be made simultaneously. However, if the point is moving, simultaneous measurements are necessary unless the magnitude of the motion over the measurement interval is insignificant in relation to the system errors (Figure 3).
GPS can also be used in this "Doppler" mode, but this is rarely done for positioning, for either navigation or surveying applications.
Figure 3. Range-differences to a single satellite for a moving antenna.
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© Chris Rizos, SNAP-UNSW, 1999