Accuracy/Computation Performance of a New Trilateration Scheme for GPS-Style Localization

We recently introduced a new geometric trilateration
(GT) method for GPS-style positioning. Preliminary singlepoint
analysis using simplistic error assumptions indicates that
the new scheme delivers almost indistinguishable localization
accuracy as the traditional Newton-Raphson (NR) approach.
Also, the same computation procedure can be used to perform
high-accuracy relative positioning between a reference vehicle
and an arbitrary number of target vehicles. This scheme has the
potential to enable a) new mission concepts in collaborative
science, b) in-situ navigation services for human Mars missions,
and c) lower cost and faster acquisition of GPS signals for
consumer-grade GPS products.
The new GT scheme differs from the NR scheme as follows:
1. The new scheme is derived from Pythagoras
Theorem, whereas the NR method is based on the
principle of linear regression.
2. The NR method uses the absolute locations (xi, yi, zi)’s
of the GPS satellites as input to each step of the
localization computation. The GT method uses the
Directional Cosines Ui’s from Earth’s center to the
GPS satellite Si.
3. Both the NR method and the GT method iterate to
converge to a localized solution. In each iteration
step, multiple matrix operations are performed. The
NR method constructs a different matrix in each
iterative step, thus requires performing a new set of
matrix operations in each step. The GT scheme uses
the same matrix in each iteration, thus requiring
computing the matrix operations only once for all
subsequent iterations.
In this paper, we perform an in-depth comparison between the
GT scheme and the NR method in terms of a) GPS localization
accuracy in the GPS operation environment, b) its sensitivity
with respect to systematic errors and random errors, and c)
computation load required to converge to a localization solution.