\section{Overall navsystem design process}\vspace{1ex}%\noindent
Modern  navigation aids are highly sophisticated systems commonly
implemented as a self-contained (autonomous) onboard inertial
navigation system (INS) aided  by many external measuring devices.
Typical external information sources would include: (1)
\textsl{posi\-tion data} -- radar, radio navigation aids, Global
Positioning System (GPS) and some others; (2) \textsl{altitude
data} -- barometric radar, radar altimeter and laser altimeter;
and (3)  \textsl{velocity data} -- Doppler radar and airspeed
sensors. Thus, the ultimate object for design is a Combined
Navigational System  including an INS aided by external
information sources (EISs). The design process of it consists of
the following 10 steps.

\textsl{Step 1}. Construct  or choose or revise a candidate
(trial) INS  \cite{broxmeyer:1956, andreev:1966,
bromberg:1979}.

\textsl{Step 2}. Choose or revise a set of EISs that will be used
in a combination with the INS.

\textsl{Step 3}. Given  the INS for analysis and the EISs, combine
available information in an optimal manner if possible. The
``optimal'' combination implies a Kalman filter in order to
generate estimates of the INS errors and to feed the estimates
back into the INS to correct it. Thus, for correction to be done
best, a Kalman filter (KF) must be designed. Clearly the KF must
be implemented using numerically stable algorithms, -- they are
widely presented in the literature \cite{kailath:1974,
morf:1975, bierman:1977, maybeck:1979, hsieh:1993, park:1995}.

The design of  an effective operational KF entails its own
iterative procedure of proposing alternative KF designs through
physical insights, tuning each, and trading off performance
capabilities and costs. A \textsl{KF systematic design procedure}
would be conducted as the following subsequence of 9 items
\cite{maybeck:1979}:

\quad\textsl{1}. Develop  a ``truth model'' that must be a complex
and as complete mathematical description of the true behavior of
the DS as possible. (Here DS stands for ``Data Source'' and
denotes the complete set of data external toward the KF, viz. the
trial INS and its EISs).

Developing  a state space ``truth model'' upon which to base the
filter is the crux of the KF design problem. The added
complication is that we must include into consideration a number
of model parameters that are uncertain for a diversity of reasons.
In fact, we must distinguish between two kinds of uncertain
parameters, -- constant values to be estimated and random noises
to be filtered. Therefore, the KF must combine filtering
state/random variables and estimating parameters. To estimate
constant parameters, many algorithms are available
\cite{kashyap:1970, kaufman:1972, ljung:1979, bierman:1990}.
Nonlinearity of the model in the values to be estimated indicates
a linearization or transition to a augmented linear model
\cite{takata:1979}.

\quad\textsl{2}.  Generate  the KF based upon the ``truth model''
as a ``benchmark'' of performance analysis and analyze its
capabilities.

\quad\textsl{3}.  Propose simplified, reduced order system models
by removing and combining states and parameters associated with
nondominant effects, deleting weak cross-coupling terms, employing
approximations, replacing auto-correlated random processes by
white discrete-time sequences and assuming Gaussian distribution
for them. This requires substantial physical insights into the
problem at hand. As a result, a set of simplified models (called
``design models'') should be created.

\quad\textsl{4}.  Develop  the simplified Kalman filters based on
each design model; also consider approximated filter structures
and possible filter adaptation mechanisms included in order to
compensate the model uncertainties that remain neglected and thus
to ensure convergence of the estimates.

\quad\textsl{5}.  Develop a structure and a program of simulated
tests. Note that the KF design procedure being described requires
high performance computational modeling tools. MathCAD, MatLAB and
Maple are examples of general purpose applications placing at our
disposal a collection of useful tools. However, the complexity of
the problem at hand does not allow us to use the standard
procedures and tool boxes, -- with them it is difficult to analyze
similar problems even of small dimensions. The reason lies in the
amount of time necessary for computing while analyzing the
influence of a set of many factors on the algorithm performance.
This raises the problem of developing a special tool box
\cite{gorokhov:2003} for investigation the complex problem
that has been described in the above items and also includes some
modules described in \cite{semoushin:2004}, viz. $(a)$
Vehicle Motion Model -- to generate a set of simulated flights for
a given type of aircraft for which an experimental INS is to be
produced on the basis of the trial INS, and $(b)$ Kinematical INS
Model -- to simulate behavior of the trial INS as a given
electro-mechanical system.

\quad\textsl{6}.  Generate a Monte Carlo analysis of designs from
Item 4 that show most promise in different simulation conditions
close to reality.

\quad\textsl{7}.  Conduct a performance-to-computer loading
tradeoff analysis and select a design; investigate different
(numerically stable) forms of implementing the chosen design
(mentioned in the start of Step 3).

\quad\textsl{8}.  Implement  the chosen design on the online
computer to be used in the final system.

\quad\textsl{9}.  Perform  checkout, final tuning, and operational
test of the filter.

\textsl{Step 4}. Schedule  and conduct simulated test flights for
the test INS. Use for scheduling a recommended set of desired
take-off flights and desired flight paths.

\textsl{Step 5}. Analyze  the simulated test flights results.
Establish the existence of (and reasons for) the construction
flaws (if any)  of the trial INS. Save those flight paths for
which the flaws manifested themselves most rapidly and patently.
Mark those flight paths as desired. Go to Step 6 if the existence
has been established, go to Step 7 otherwise.

\textsl{Step 6}. Decide  whether the reasons established at Step 5
lie in the DS modeling or not. If ``Yes'', repeat Step 3 starting
it by refining the DS ``truth model''. If ``No'', go to Step 1.

\textsl{Step 7}. The  experimental INS is produced and bench
tested. Schedule and conduct field test flights for the
experimental INS on board. Use for scheduling the set of desired
flight paths saved at Step 5.

\textsl{Step 8}. Analyze the field test flights results. Establish
the existence of (and reasons for) the construction flaws (if any)
of the  experimental INS. Go to Step 9 if the existence has been
established, go to Step 10 otherwise.

\textsl{Step 9}.  Decide  whether the reasons established at Step
8 lie in the Step 7, Step 4, Step 3, Step 2, or Step 1.
Repeat the steps needing revision.

\textsl{Step 10}. Acceptance  test. Rethink the problem and revise
what needs revision (or re-work).
