Boots – shoes – and leggings
Patent
1994-11-01
1997-08-05
Cosimano, Edward R.
Boots, shoes, and leggings
36457101, G06F 1900
Patent
active
056549075
DESCRIPTION:
BRIEF SUMMARY
TECHNICAL FIELD
This invention relates generally to all practical applications of the Kalman Filter and more particularly to large dynamical systems with a special need for fast, computationally stable and accurate results.
BACKGROUND ART
Prior to explaining the invention, it will be helpful to first understand the prior art of both the Kalman Filter (KF) and the Fast Kalman Filter (FKF.TM.) for calibrating a sensor system (WO 90/13794). The underlying Markov process is described by the equations from (1) to (3). The first equation tells how a measurement vector y.sub.t depends on the state vector s.sub.t at timepoint t, (t=0,1,2 . . . ). This is the linearized Measurement (or observation) equation: of the actual Measurement equations. The second equation describes the time evolution of e.g. a weather balloon flight and is the System (or state) equation: s.sub.t-1 as well as of increments u.sub.t-1 and a.sub.t These increments are typically caused by a known uniform motion and an unknown random acceleration, respectively.
The measurement errors, the acceleration term and the previous position usually are mutually uncorrelated and are briefly described here by the following covariance matrices: -s.sub.t-1)'}(3)
The Kalman forward recursion formulae give us the best linear unbiased estimates of the present state +u.sub.t-1)} (4) R.sub.a.sbsb.t)H'.sub.t +R.sub.e.sbsb.t }.sup.-1 ( 6)
Let us now partition the estimated state vector s.sub.t and its covariance matrix P.sub.t as follows: ##EQU1## where b.sub.t tells us the estimated balloon position; and, follows: ##EQU2##
The recursion formulae from (4) to (6) gives us now a filtered (based on updated calibration parameters) position vector (s.sub.t-1 +u.sub.t-1)} (9) (st-1+u.sub.t-1)} (10) ##EQU3##
The following modified form of the general State equation is introduced the Measurement equation (1) in order to obtain so-called Augmented Model: ##EQU4## The state parameters can now be computed by using the well-known solution of a Regression Analysis problem given below. Use it for Updating: z.sub.t ( 14) not numerically. For the balloon tracking problem with a large number sensors with slipping calibration the matrix to be inverted in equations (6) or (11) is larger than that in formula (14).
The initialization of the large Fast Kalman Filter (FKF.TM.) for solving the calibration problem of the balloon tracking sensors is done by Lange's High-pass Filter. It exploits an analytical sparse-matrix inversion formula (Lange, 1988a) for solving regression models with the following so-called Canonical Block-angular matrix structure: ##EQU5## This is a matrix representation of the Measurement equation of an entire windfinding intercomparison experiment or one balloon flight. The vectors b.sub.1,b.sub.2, . . . ,b.sub.K typically refer to consecutive position coordinates of a weather balloon but may also contain those calibration parameters that have a significant time or space variation. The vector c refers to the other calibration parameters that are constant over the sampling period.
The Regression Analytical approach of the Fast Kalman Filtering (FKF.TM.) for updating the state parameters including the calibration drifts in particular, is based on the same block-angular matrix structure as in equation (15). The optimal estimates ( ) of b.sub.1,b.sub.2, . . . b.sub.K and c are obtained by making the following logical insertions into formula (15) for each timepoint t, t=1,2, . . . : ##EQU6## These insertions concluded the specification of the Fast Kalman Filter (FKF.TM.) algorithm for calibrating the upper-air wind tracking system. Another application would be the Global Observing System of the World Weather Watch. Here, the vector y.sub.k contains various observed inconsistencies and systematic errors of weather reports (e.g. mean day-night differences of pressure values which should be about zero) from a radiosonde system k or from a homogeneous cluster k of radiosonde stations of a country (Lange, 1988a/b). The calibration drift vector b.sub.k will then tell us wh
REFERENCES:
patent: 5506794 (1996-04-01), Lange
Cosimano Edward R.
Smith Demetra R.
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