RECEIVER-BIASED RANGE POSITIONING |
The observation equation for a receiver-biased range is (eqn
(1.3-14)):
*(t) = (t) + d(t_{r}).c |
where c is the speed of electromagnetic radiation, d is the receiver clock error caused by the receiver oscillator (satellite time is assumed to be "true" time), * is the measured range and is the true range. Each observation made by the receiver can be parameterised as:
(x^{s} – x)^{2} + (y^{s} – y)^{2} + (z^{s} – z)^{2} = (* – d.c)^{2} | (1.4-1) |
where the time argument has been discarded.
If it is assumed that the coordinates of the transmitter (land-based or satellite-based) (x^{s}, y^{s}, z^{s}) are known, then each measurement * contains four parameters which are unknown: the 3-D coordinates of the receiver (x, y, z) and the receiver clock error (). If four measurements are made, to four different targets, the following system of equations can be constructed:
(x^{s1} – x)^{2} + (y^{s1} – y)^{2} + (z^{s1} – z)^{2} = (* ^{1}– d.c)^{2} | |
(x^{s2} – x)^{2} + (y^{s2} – y)^{2} + (z^{s2} – z)^{2} = (* ^{2}– d.c)^{2} | |
(x^{s3} – x)^{2} + (y^{s3} – y)^{2} + (z^{s3} – z)^{2} = (* ^{3}– d.c)^{2} | |
(x^{s4} – x)^{2} + (y^{s4} – y)^{2} + (z^{s4} – z)^{2} = (* ^{4}– d.c)^{2} | (1.4-2) |
This system of equations has a unique solution. If more than four measurements are made, the method of Least Squares can be used to obtain the optimal solution. Least Squares provides, in addition to the solution for the unknown parameters, an estimate of the quality of the positioning solution.
The steps in a Least Squares solution generally are:
(1) Set up the solution: compute the elements of the design matrix A , containing the partial derivatives of the range observations with respect to the parameters:
(1.4-3) |
(2) Obtain approximate (or apriori) estimates of the parameters: in particular the geodetic parameters to be used for the computation of the partial derivatives and the residual quantities (the sum of squares of which are to be minimised):
= ( - () ) | (1.4-4) |
(3) Specify the quality of the observations: by defining the weight matrix P.
(4) Form the normal matrix: N = A^{T}PA , and solve the system of equations:
= (N)^{-1} A^{T}P | (1.4-5) |
This is the standard mode of pseudo-range positioning used in GPS navigation, in which the receiver clock error is treated as an additional unknown. All other biases are assumed to be insignificant (that is, their impact on the quality of the position solution is considered negligible). The GPS satellite clock error can be considered a known quantity, and parameters correcting this bias are transmitted in the Navigation Message (section 3.3.2).
Would it be necessary to solve for the receiver clock error at each epoch? That would depend upon:
A study of the Allan Variance graph, and assuming: (a) a range measurement precision of about 1 metre due to "noise", and (b) the receiver is equipped with a quartz clock; then the unpredictability of the clock after 30 seconds is as great as the uncertainty of the range measurement. Clearly, once the clock error was determined, it would have to be independently estimated at least every 30 seconds otherwise it would dominate the "equivalent range error". This is one of the reasons why this method could not be used if the measurements were not made "simultaneously" on all transmitters (here, within 30 seconds of each other, so that the clock error can be assumed a constant). In fact the receiver clock can be "reset" to its "true" time on a regular basis, so that the drift of the clock (and the consequent contamination of the range measurements) can be constrained, as shown schematically in Figure below.
Figure 1. Clock drift and periodic reset.
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© Chris Rizos, SNAP-UNSW, 1999