# COMPUTATION nature of the filter itself. Seldom do

COMPUTATION OF EXTENDED ROBUST KALMAN FILTER FOR REAL-TIME
ATTITUDE AND POSITION ESTIMATION

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Ghulam
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Abstract:

This paper
explains the implementation of extended robust Kalman filter. This demonstrates
a method of real time computation of the filter on parallel computing devices.
The solution of the filter expressed as a set of simultaneous linear equations
which can be evaluated using givens rotation. The paper also represents ERKF in
the development of attitude and position reference system for cargo transport
vehicle. This paper concludes by analyzing the performance of the ERKF and
verifying the validity of the givens rotation method.

Kalman Filter:

Inside the huge tool compartment of scientific tools that
can be utilized for stochastic estimation from noisy sensor estimations, a
standout amongst the most surely understood and frequently utilized tools is
what is known as the Kalman filter. The Kalman filter is named after Rudolph E.
Kalman, who in 1960 published his well known paper depicting a recursive answer
for the discrete-data linear filtering problem.

The Kalman filter is basically an arrangement of equations
that implement an optimal estimator that reduces the estimated error at
maximum. The Kalman filter has been the subject of broad research and application,
especially in the area of navigation. This is likely due in extensive part to
advances in digital computing that made the utilization of the filter in real
time practical problems, yet additionally to the relative simplicity and robust
nature of the filter itself. Seldom do the conditions essential for optimality
really exist, but then the filter obviously functions admirably in spite of
situations, for some applications despite this circumstance.

The Kalman
filter is to estimate the state of a discrete-time controlled process which is governed by the linear
stochastic difference equation

With measurement  which
is

The random variables wk and Vk represent the process and
measurement noise respectively.

Extended Kalman Filter (EKF):

A Kalman filter that linearizes itself about the mean
and covariance is said to be as
Extended Kalman Filter or EKF. Calculations, in this specific
paper were made using provided rotations. While talking about vehicle state
determination the system was composed of inertial measurement units and GPS.
IMU was using uncertain results of rates of accelerometers and gyros.

The dynamic equations for attitude model are given by

and  are
roll, pitch and yaw respectively.

Transformation
matrix between angular velocities is

Recursive algorithm used in the paper is shown in below.

Experimental Results:

IMU and GPS were used to
find position (lat, long, alt) and attitude (roll, pitch, heading) of a
vehicle. Micro electro mechanical system based sensors were used. The IMU update
frequency was 400Hz having a built-in algorithm to compute linear and angular
velocities and orientations by sensors. It used gyros, accelerometers and
magnetometers. Velocities (angular and linear) were measured about the vehicle
frame of reference and orientation was observed in terms of Euler angles. The
GPS provided the values of yaw, roll, pitch, lat, long and alt. Its frequence
was 10Hz which was far less than that of IMU which was 400Hz.

Attitude Estimation:

As we can see in the
figures below, variations in roll and pitch are considerably acceptable while
that in heading are higher which was exactly as expected for ground vehicle.

Roll                        Pitch

Moreover it was observed
that GPS measurements were more reliable and certain in this specific case than
those of inertial measurement unit which were have more uncertainty. Further,
because of extended robust Kalman filter the estimations of heading were robust
to uncertainties in the IMU measurements.

Position Estimation:

Estimation of position was
fitted with GPS readings. Further, when due to unavailability of GPS, inertial
navigation systems were measured, it used the frequency of 400Hz instead of
10Hz, used in the case of GPS. The extended robust Kalman filter was
implemented by the approach of inversion of conventional matrix as well as that
of proposed given rotation. The respective singular values (max and min) of the
position system matrix were also discussed.

Position:         Latitude                                  Longitude
Altitude

Conclusion:

In this paper attitude and
yaw reference system was discussed and presented which was based on inertial
measurement unit and GPS data using extended robust Kalman filter. The
estimation of heading, given by the filter was according to the accurate GPS
measurements. The estimations of position were made at higher frequencies due
to the use of INS based on the prediction of attitude of the body. And the
filter made it sure that the predictions were robust to uncertainties in both
models. Moreover a method of computing the extended Kalman filter in real time
was presented and cross verified which was based on QR decomposition. Further
it was discussed that this method had increased computational effectiveness
over conventional approach.

References:

1-
Bijker, J. and Steyn, W. (2008).
Kalman filter configurations for a low-cost loosely integrated inertial
navigation system on an airship, Control Engineering
Practice 16(12): 1509-1518

2-
Golub, G. H. and Loan, C. F. V.
(1996). Matrix Computations, The Johns Hopkins University Press.

3-      Robust Recursive Kalman Filtering for Attitude Estimation by Roberto
S. Inoue ? Marco H. Terra (September2, 2011)

4-      A Robust Extended Kalman Filter for
Discrete-time Systems with Uncertain
Dynamics, Measurements and Correlated Noise Rodrigo Fontes Souto, Student
Member, IEEE, Joa˜o Yoshiyuki Ishihara, Member, IEEE, Geovany Arau´jo Borges, Member, IEEE (June 10-12, 2009

5-      Nonlinear Dynamic System Identification Based on Fuzzy
Kalman Filter Dan´ ubia Soares Pires and Ginalber Luiz de Oliveira Serra Federal
Institute of Education, Science and Technology Department of Electroelectronics Laboratory
of Computational Intelligence Applied to Technology S˜
ao Luis – MA, Brazil

6-
A New
Approach to Linear Filtering
and Prediction Problems by R.E. Kalman Research Institute for Advanced Study,2 Baltimore,
Md. (1960)

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