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





sin
cos
V (3)
Here GS is the recorded groundspeed and  is the true track angle. Sometimes the track angle has not been measured
and has to be computed from the true heading angle t and the dri ft angle that have been measured. If also the dri ft
angle has not been measured, which sometimes occurs, the wind estimation process breaks down. In this exceptional
case, in order to salvage the wind calculational process an estimate of the dri ft angle is obtained using the FMSrecorded
wind speed and direction, together with other available inertial and aerodynamic data. The FMS-wind is
normally computed from drift angle, track angle, heading and inertial and airspeed, so the drift angle is calculated
using this computational process in reverse. The FMS-wind information, however, does not contain a vertical wind
component and may have other dynamical errors (e.g. the aerodynamic sideslip angle is neglected) and/or time lags.
In subtracting the velocities in Eq.(1) they have to be referenced in the same reference frame. The transformation
from the B-frame to the E-frame and vice-vers a is done through the transformation matrix Tbe, so that for example
  e
be
Vb  T V
This transformation matrix contains the well-known axis transformation expressions involving the Euler angles
,  and . In case of reverse transformation one gets     b
be
e b
V Teb V T 1V    , where one can prove that
   T
Tbe  Tbe 1 .
4
An important contribution to the computation of the wind vector is the vertical inertial velocity z . It contributes
to a significant extent to the determination of the vertical wind component and is therefore important to be estimated
as accurately as possible.
2.3Kalman filtering and smoothing
A main feature of the data analysis software is the application ofKalman filtering and smoothing. It is a process
of estimating the state vector of a dynamical system at a particular stage i (or time ti) and its covariance by using the
measurements at all stages. The Kalman filter-smoother in the present algorithm is used speci fically to estimate the
inertial vertical speed as accurately as possible, which is an element of the state vector x that is estimated, consisting
of 3 velocities, 3 positions and 3 accelerometer biases. Measurements used are inertial data (e.g. track,
groundspeed), attitudes (Euler angles), dri ft angle, baro and radio altitudes, etc.
2.3.1 Filtering pass
In the filtering pass through time the state vector x of the dynamical system is estimated using measurements,
taken at 1-sec time intervals on average. The filtering process runs through a prediction-measurements-update cycle
in discrete time, or stages, as follows:
prediction from stage i-1 to i:
xi  Φi1ˆxi1  Bi1ui1  Γi1wi1
measurements taken at stage i:
zi  Hixi  υi
update the predicted state:
xˆi  xi  Ki zi Hixi 
Generally 4 prediction cycles (at 0.25s) are run, followed by one update cycle (per second).
The “ control” inputs u are the body accelerometer signals; w is the acceleromet er measurement noise. The matrix Ki
is the well-known Ricatti matrix.
2.3.2 Smoothing pass
After the filtering process is completed, the smoothing process starts. The smoothed results are the best estimate
of the state vector x at time i, given all the measurements over the entire interval, 1-N.
The state is smoothed using
i
Ti
xˆ i/N  xˆ i  PiΦ λ
where the co-st ate variable 
i is obtained from the smoothing process, which runs backwards in time:
λ   I K H  Φ λ  H Ri zi  Hixi  λ N  0
Ti
i
Ti
T
i i i with 1
1
Also the covariance matrix Pi is updated to Pi/N. More details can be found in Ref. 2.
The Kalman filtering process has been quite generally used in many applications. The Kalman smoother,
however, has not found wide application, mostly also because it can only be applied in a post-processing mode, i.e.
after all data has been taken and process ed forward in time. If a “ real-time” estimator is to be implemented then only
the Kalman filter is applicable.
The increased accuracy in the estimate of altitude at lower altitudes due to the radio altimeter being used helps in
　
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