Hi : )
today a very short entry about filtering. Most of information will be given
from external sources.
At first Kalman filter:
 |
| Fig. 1. Principle of Kalman operation |
It is very
sophisticated method for estimation of measured values. But, it works
perfectly, like no other filter. Every kind of measurement has some accuracy
error; additionally during measure occurs some noisy (mainly in electronic
measuring equipment). We can mention that also our measurement device perturb
measurement process. All of those factors cause inaccuracy of our measurement.
We have to deal with (sometimes) huge differences between real and measure value.
If we want to minimize this factor, we have to use some
estimation/approximation methods, so-called filtering techniques. One of them
is Kalman filter. It is so-called linear
quadratic estimation (LQE), and an algorithm uses a series of measurements
observed over time, containing noise (random variations) and other
inaccuracies, and produces estimates of unknown variables that tend to be more
precise than those based on a single measurement alone. More formally, the
Kalman filter operates recursively on streams of noisy input data to produce a
statistically optimal estimate of the underlying system state. The filter is
named for Rudolf (Rudy) E. Kálmán, one of the primary developers of its theory.
All of
theory of Kalman filter is very mathematically complicated, and till now we don’t
understand it : ) but we will work at it ; ) more information in references.
Complementary
filter
Complementary
filter bases in simplest way of two filters connected in parallel: low-pass
filter and high-pass filter:
 |
| Fig. 2. Principle of Complementary filter |
Mathematical
description is as following: The complementary filter is a frequency domain
filter. In its strictest sense, the definition of a complementary filter refers
to the use of two or more transfer functions, which are mathematical complements of one another. Thus, if the data
from one sensor is operated on by G(s), then the data from the other sensor is
operated on by I-G(s), and the sum of the transfer functions is I, the identity
matrix. In the case of a one-dimensional filter as will be described in this
paper, the identity matrix reduces to the scalar number
one.
In a
typical two -input system, one input will provide information with high
frequency noise, and is thus low -pass filtered. The other input provides
information with low frequency noise, and is high-pass filtered. If the low
-pass and high-pass filters are mathematical complements, then the output of
the filter is the complete reconstruction of the variable being sensed, minus
the noise associated with the sensors.
Unfortunately,
theory of complementary filters is not described well; so I have to reference
into some Internet links.
However,
how co connect measurements from gyroscope and accelerometer, and then try to
find original (real) value?
At first,
we have to assume as following:
Then, we
have to think how to estimate our values (now, more or less, they are in the
same units ; ) )
We can use
here some model to interpret such a situation:
It is
possible to implement Kalman filter as following:
But it
needs more time to spend on it. Much easier to implement complementary filter:
And short explanation:
At the end,
our results. We implement exactly like above, complementary filter (the same
coefficient, at the beginning). Results are shown below. In first chart –
accelerometer measurement, below gyroscope and at the bottom – filtered values.
It’s everything for today : )
References:
1. http://www.cs.unc.edu/~welch/kalman/
2. http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx
3. http://www.glassercommunications.com/paul/samples/filters_for_fusion.pdf
4. http://robottini.altervista.org/kalman-filter-vs-complementary-filter
And the main source:
web.mit.edu/scolton/www/filter.pdf
