"What are the different roles that Helmert, Wolf, K- and Lange played in the development of FKF?"

The answer:

"It has turned out that:

In 1880, Professor R. F. Helmert (1841-1917) specified the underlying mathematical problem
through formulating the sparse Canonical Block-Angular (CBA)
equation system
for GEODETIC data.

In 1960, the equation system and formulas for solving positions of MOVING objects were
disclosed under the title of K- Filtering
(KF).

In 1969, a computing method for
solving COMMON factors and (Canonical) Correlations between (two) blocks of data by using
Empirical Orthogonal Functions (EOF) was reported by
Lange (1969).

In 1972-75,
the sparse CBA equations
for balloon tracking systems were
reported
and
solved
ANALYTICALLY by Lange.

In 1978, Professor Helmut Wolf (1910-1994) published his analytic
formulas
for a blockwise solution of Helmert's Normal Equations
(NEQ).

In 1982, an exact formula for computing the
ACCURACY of the Helmert-Wolf blocking (HWB) solution
was published by
Lange (1982).

In 1986, the HWB solution was GENERALIZED to other
applications by
Lange (1987).

In 1989, the CONNECTION between the HWB solution
and fast K- Filtering (FKF) was discovered and patented by
Lange (1990).

In 1992, the FKF method was applied to EXTENDED K- Filtering (EKF) by
Lange (1993).

In 1995, the FKF method was extended to ADAPTIVE K- Filtering (AKF) by
Lange (1997). "

"Could the FKF method be replaced by using much faster microprocessors?"

The answer:

"No, only theoretically. The FKF method may though be circumvented by using a furiosly fast
processor. However, this would take place at the expense of a serious increase in power consumption.
The battery of a light-weight low-powered device
should be many times larger for equal high-performance-computing (HPC) without the use of FKF.

FKF is expected to become a killer application when serious
competition in both accuracy and reliability begins also
in the civilian markets of precision navigation, mobile positioning, ultra-reliable guidance, etc."

"How do you explain the very high navigational accuracies expected?"

The answer:

"FKF renders the most effective computational method for updating
calibration parameters that stem as integration constants
from the differential equations of signal-phases and acceleration
measurements in HYBRID systems.
The optimality of the FKF computations is necessary for reliable
accuracy estimation that can now be obtained as a byproduct from
the sophisticated theory of Minimum Norm Quadratic Unbiased Estimation
(MINQUE) by C. R. Rao (1975).
Superb accuracies and integrity are thus achieved in real-time only by using the
Statistical Calibration
by Lange (1999)."

"How will the very high accuracies be achieved?"

The answer:

"The International GPS Service
(IGS) develops its orbital
solutions towards realtime application.
The HWB solution will then need
simultaneous results
from meteorological data-assimilation like that of
the European Centre for Medium-Range Weather Forecasts
(ECMWF).
Embedded microchips based on hybrid navigation concepts are intensively developed.
The FKF patents offer industrial protection to the big R&D efforts and investments required."

"Who holds these FKF patents?"

The answer:

"Several small European companies, please see:
Intellectual Property "

"How to explain the K-Filter (KF) in simple terms?"

The answer:

"KF renders the theoretically best method for updating
estimates of unknown parameters when new data flow
continuously into a navigation receiver or control system.
Thus, all navigation receivers must make use of a K-Filter.
However, there still are certain necessary conditions for
these filtering processes to be absolutely RELIABLE.

All roads may lead to Rome, however, the consequencies can be dead
serious if one hits something else before there. "

"How is your FKF related to the KF?"

The answer:

"One of the reliability conditions of a KF
is that the continuously inflowing
data must contain enough information on those
parameters whose values must be estimated. In other words,
if a KF tries to estimate the value of a parameter under circumstances
when the inflowing data has very little to do with the parameter
then the estimated value is doomed to go astray sooner or later.
The FKF-method represents the Best Available Technology (BAT) of
extracting such information from the inflowing data.

FKF cannot make miracles but it will
save lives because of the ever-increasing automation that
is relied upon.
This opportunity for improving and warranting public safety must not be deferred
by ignorance and excuses."

"Can you give some ideas of the speed of the computations with your FKF technique compared to the UD implementation?

Are already libraries implemented ready for use?"

The answer:

"The FKF and UD techniques are by no means exclusive. The semi-analytical computations of FKF
offer significant advantages over UD:

- Large moving batches of data can be submitted for a careful Minimum Least Squares analysis in order to detect and remove calibration drifts; and,

- Operational accuracies of an overdetermined navigation receiver can be reliably estimated
in realtime from observed internal consistencies of the signals and sensors actually in use.

In contrast, if the data are processed one at a time by a typical UD (Square-Root) filter then
crucial information on weakly observable calibration parameters can easily be lost.
This may lead to serious consequencies due to an unobserved filter divergence.

Sample subroutines of FORTRAN77 can be found in CALLIB.for
though without any documentation as different applications require largely varying pieces of
such code."

"Could you please give a simple example of how FKF is used?"

The answer:

"My PhD
thesis
describes a prototype system for the tracking of weather balloons.
For additional information, please, just let me know:
Lange@FKF.net."

"Does this mean that simpler weather forecasting is on the way? :-)

The answer:

"Certainly! We shall soon be able to buy 'super'-navigation receivers
attached to mobile phones. These will provide positional information
and predict the weather at the same time.
In fact,
FKF offers the best possible ratio between
navigational accuracy and battery saving.
However, FKF will give no shortcut to the El Nino problem because of
its global nature. The Global Circulation Models (GCM) of atmosphere
and oceans are probably too sophisticated to be run on a personal
computer"

"I cannot understand how this relates to the Christian faith."

The answer:

"We all are God's laborers and Lord Jesus Christ is waiting for
His bride to emerge soon. All those individual persons are His bride who
love Him as a good wife loves her husband. The labor means taking
part in His sufferings that everyone must face when helping people
out from spiritual darkness to light.

There are many ways of serving the Lord. An important way is to
sponsor ministers to preach the Good News of Him. However,
the people of many countries are so distressed that they need
help themselves. A way of helping them is to offer well-payed work.
However, no company can afford paying good salaries if it cannot
sell its products or services at reasonable prices. The products must be of a very good
quality or of high technology when made of cheap raw materials. Thus, the way to
prosperity is to know how to make good products in quantities and to possess
the licencies to sell them to a large market.

The Lord gave me the invention of the Fast K-Filter (FKF) in sleep
in order that we together can help His own people by granting FKF
lisencies. The Lord did a similar thing in my country by sending
James Finlayson of Scotland to initiate the Industrial Revolution
of Finland already in year 1817."

It was asked:

"How to use FKF for solving the following
typical problem:

X(k+1)=A(k)X(k)+W(k)

Y(k)=H(k)X(k)+V(k) (k is time)

where X is the state vector and Y is the observation vector. With matrices A and H known, how to get vector X at each time using the FKF program modules?"

The answer:

"Your equation system is valid for prediction, not for recursive KF
estimation.
At first, you should write it for KF e.g. as follows:

X(k)=A(k)X(k-1)+W(k)

Y(k)=H(k)X(k)+V(k) (k is time)

Then subtract A(k)Z(k-1) from both sides of the first equation (Harvey's approach) in order to get the so-called Augmented Model:

A(k)Z(k-1)=X(k)+A(k)(Z(k-1)-X(k-1))-W(k)

Y(k)=H(k)X(k)+V(k)

where Z(k-1) is an estimate of X(k-1) from the previous time step from k-2 to k-1.

Now, the FKF program modules apply for computing the maximum likelihood estimate Z(k) of X(k) from the time step from k-1 to k. Please observe that the Augmented Model is just a typical linear regression problem that can be solved by using any Least Squares Analysis software package. FKF speeds up the computations enormously in many cases where vectors Y(k) and/or X(k) have many components."

* Last revised: December 9, 2003