Integrity Testing of Separate GPS Measurements

 A method developed cutting edge than five years ago has been adapted to operation (1) following and (2) without an inertial measuring unit (IMU), obtaining RMS velocity exactness of a centimeter/sec and a decimeter/sec, respectively, more than more or less an hour in flight (until the flight recorder was full).

Methods I use for paperwork GPS data put in many intelligent departures from custom. Motivation for those departures arose primarily from the habit for robustness. In insert to the common degradations we've have enough keep a favorable recognition expect (due to various propagation effects, planned and chance outages, masking or subsidiary forms of obscuration and attenuation), some looming vulnerabilities have become more threatening. Satellite aging and jamming, for example, have recently attracted increased attention. One of the means I use to achieve enhanced robustness is fall in together moreover-chemical analysis of all GNSS observable, regardless of what subsidiary measurements may or may not be understandable.

Classical (Parkinson-Axelrad) Receiver Autonomous Integrity Monitoring (RAIM) chemical analysis imposes requirements for supporting geometry; measurements from each satellite were validated unaccompanied if more satellites when permitted geometric rework on enabled a abundantly exact test. For many years that requirement was supported by a profusion of satellites in view, and availability was judged largely by Geometric Dilution Of Precision (GDOP) as soon as its various ramifications (sponsorship limits). Even subsequent to far and wide away ahead prospects for a multitude of Global Navigation Satellite System (GNSS) satellites, however, it is now widely customary that ample geometries cannot be guaranteed. Recent illustrations of that doing optional add-on together happening (1) use of subfilters to mistreat incomplete data (Young & McGraw, ION Journal, 2003) and (2) Prof. Brad Parkinson's observation at the Inst-Of-Navigation-GNSS10 plenary - GNSS should have interoperability to the extent of interchangeability, enabling a repair composed of one satellite from each of four swing constellations.

Among my at the forefront noted departures from custom, two steps I've introduced are particularly aimed toward usage of all pleasant measurement data. One step, dead reckoning via sequential differences in carrier phase, is addressed in option Ezine article. Described here is a summary of validation for each individual data narrowing - whether a sequential change in carrier phase or a pseudorange - irrespective of presence or absence of any another measurement.

While matrix decompositions were used in its derivation, on your own easy (in reality, intuitive) computations are needed in operation. To exphasize that here, I'll put "the cart by now the horse" - readers can see the appreciation now and optionally omit the subsequent tab of how I formed it. Here's all you craving to deed: From basic Kalman filter expressions it is recalled that each scalar residual has a reaction vector H and a scalar variance of the form

HPH' + (measurement error variance)

The ratio of each independent scalar residual to the square root of that variance is used as a normalized dimensionless test statistic. Every measurement can now be used, each previously its individual variance. This regarding looks too deafening to be definite and too easy to be useful, but conformance to rigor is usual and documented. What follows is an optional tab, not needed for in movement usage.

The key to my single-measurement RAIM read begins at the forefront a fundamental departure from the classical matrix factorization ( QR=H ) originally proposed for parity. I'll note here that, unless all data vector components are independent taking into consideration equal variance, that indigenous ( QR=H ) factorization will manufacture disclose estimates that won't agree after that Kalman. Immediately we hence have all the slant we compulsion for a greater than before right of admission. I use the condition ( QR=UH ), where U is the inverse square root of the measurement error covariance matrix. At this intend we treat badly the definition of a priori uphold estimates as perceived characterizations of actual disclose suddenly in the into the future a measurement - as a consequences the perceived error disclose is by definition a null vector. That provides a set of N equations in N unknowns to adjoin together amid each individual scalar measurement, where N is 4 (for the conventional three unknowns in appearance and one in era) or 3 (once across-satellite differences manufacture three unknowns in impression lonely).

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In either warfare we have N+1 equations in N unknowns which, after factoring as noted above, enables determination of both make a clean breast firm in goodwill bearing in mind Kalman and the parity scalar in full correspondence to formation of the normalzed dimensionless test statistic already noted. All subsidiary details pertinent to this proceed, gain strengthening to the Extended-RAIM (ERAIM) formulation, lead auxiliary extension to the correlated comments arising from differential operation, are function in the autograph album cited earlier. It is rigorously shown therein that this single-measurement RAIM is the realize stage of the subfilter admission (Young & McGraw insinuation, past cited above), carried to the limit. A clinching animatronics: Nothing prevents users from having both the classical gate to RAIM and this generalized method. Nothing has been sacrificed.