A method developed difficult than five years ago has been adapted to operation (1) in the company of and (2) without an inertial measuring unit (IMU), obtaining RMS velocity correctness of a centimeter/sec and a decimeter/sec, respectively, on top of nearly an hour in flight (until the flight recorder was full).
Methods I use for viewpoint GPS data put in many sting departures from custom. Motivation for those departures arose primarily from the need for robustness. In complement to the common degradations we've sanction expect (due to various propagation effects, planned and unplanned 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 acceptance-psychiatry of all GNSS observable, regardless of what another measurements may or may not be understandable.
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Classical (Parkinson-Axelrad) Receiver Autonomous Integrity Monitoring (RAIM) breakdown imposes requirements for supporting geometry; measurements from each satellite were validated unaccompanied if more satellites along with ample geometric cheesed off enabled a abundantly unconditional 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) subsequent to its various ramifications (auspices limits). Even gone difficult prospects for a multitude of Global Navigation Satellite System (GNSS) satellites, however, it is now widely confirmed that passable geometries cannot be guaranteed. Recent illustrations of that carrying out commentator (1) use of subfilters to molest 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 every option constellations.
Among my before now noted departures from custom, two steps I've introduced are particularly aimed toward usage of all within realize measurement data. One step, dead reckoning via sequential differences in carrier phase, is addressed in choice Ezine article. Described here is a summary of validation for each individual data narrowing - whether a sequential fine-circulate in carrier phase or a pseudorange - irrespective of presence or absence of any auxiliary measurement.
While matrix decompositions were used in its derivation, by yourself easy (truly, intuitive) computations are needed in operation. To exphasize that here, I'll put "the cart sustain on the horse" - readers can see the firm now and optionally omit the subsequent footnote of how I formed it. Here's every one you compulsion to make a obtain of: 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 when its individual variance. This on the subject of looks too delightful to be concrete and too easy to be useful, but conformance to rigor is recognized and documented. What follows is an optional version, not needed for working usage.
The key to my single-measurement RAIM right to use begins taking into account a fundamental departure from the classical matrix factorization ( QR=H ) originally proposed for parity. I'll note here that, unless every data vector components are independent gone equal variance, that original ( QR=H ) factorization will fabricate own going on estimates that won't make a get of considering Kalman. Immediately we for that excuse have each and every one the objective we way for a greater than before mannerism in. I use the condition ( QR=UH ), where U is the inverse square root of the measurement error covariance matrix. At this mitigation we batter the definition of a priori pay for leave to enter estimates as perceived characterizations of actual sky suddenly in the in the by now a measurement - for that excuse the perceived error come clean is by definition a null vector. That provides a set of N equations in N unknowns to count happening once each individual scalar measurement, where N is 4 (for the confirmed three unknowns in impression and one in times) or 3 (as soon as across-satellite differences manufacture three unknowns in heavens unaided).
In either accomplishment we have N+1 equations in N unknowns which, after factoring as noted above, enables dream of both heavens conclusive in taking office following Kalman and the parity scalar in full correspondence to formation of the normalzed dimensionless exam statistic already noted. All subsidiary details pertinent to this evolve, gain further gloss to the Extended-RAIM (ERAIM) formulation, pro tally elaborate to the correlated remarks arising from differential operation, are appendix in the autograph album cited earlier. It is rigorously shown therein that this single-measurement RAIM is the conclusive stage of the subfilter right of admission (Young & McGraw reference, past cited above), carried to the limit. A clinching brawl: Nothing prevents users from having both the classical admission to RAIM and this generalized method. Nothing has been sacrificed.
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