Article citation information:
Banachowicz, A., Wolski, A. Methods of position
estimation in parametric navigation. Scientific
Journal of Silesian University of Technology. Series Transport. 2017, 95,
19-25. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2017.95.2.
Andrzej BANACHOWICZ[1],
Adam WOLSKI[2]
METHODS OF POSITION ESTIMATION
IN PARAMETRIC NAVIGATION
Summary. The estimation of
position coordinates of a navigating ship is one of the navigational
subprocesses. The methods used in this process are either deterministic (the
case of a minimum number of navigational parameters measurements) or
probabilistic (in cases where we have access to information redundancy).
Naturally, due to the accuracy and reliability of the calculated coordinates,
probabilistic methods should be primarily used. The article presents
the use of the method of least squares and
Kalman filtering in algorithms in integrated navigation for the estimation of
position coordinates, taking into account ship movement parameters.
Keywords:
navigational data fusion; least squares estimation; Kalman filtering;
estimation; navigation; algorithm of integrated navigation.
1. INTRODUCTION
The calculation of ship coordinates
involves measurements and calculations of various navigational parameters. The
basic navigational parameters include position coordinates and movement
(velocity vector). In terms of position determination, there are three groups
of navigation methods:
- Dead reckoning, in which, on the basis of the
mathematical model of the ship movement and velocity vector measurements, we
can determine a ship’s position at any time. Such an approach is known as dead
reckoning navigation, with inertial navigation being one of its technical
implementations.
- Comparative methods, which involve the comparison of
the physical field recorded in an analogue or digital device with its measured
values. These methods are used in bathymetric or topographic navigation, that
is, navigation based on the measurement of physical fields of the Earth
(magnetic, gravitational and other). Today, these methods are also used for
comparing radar images with digital charts.
- Parametric methods, using the measurement of physical
quantities, which directly or indirectly determine navigational function, i.e.,
geometric relations between a ship’s position and navigational marks’
coordinates. This is the primary method of position fixing.
The algorithms for integrated
navigation systems involve a fusion of different methods; in particular, the
parametric method is combined with the dead
reckoning method. This process requires the combined
processing of measurement data, which allows us to optimize the use of
navigational information. The multisensor fusion of navigational data is widely
discussed in the literature, e.g., [7], while GPS data integration with other
navigational measurements is described in [3].
These authors present
selected variants of the integration of navigational data obtained from
different navigation systems. The method of least squares and the classic
Kalman filter were used as the mathematical model of measurements integration.
2. THE LEAST SQUARES METHOD
Let us assume that we have
measurements of varying accuracy and that we will use the method of least
squares with weights for their fusion [9]. In this case, the vector of state
(position coordinates) is described by this formula:
and its covariance matrix is written
as this relation:
where:
One of the simple measurement
situations is a combination of GPS position coordinates with a dead reckoning (DR) position. In this case,
The matrix of measurements covariance will mean that
where the matrices
Here, we have used the matrix
inversion by division into blocks [8]:
where:
Ultimately, with the above
assumptions, we get the vector of state (position coordinates):
and its covariance matrix:
3. KALMAN FILTER
Kalman
filtering is commonly used today [5], [6]. It is implemented at various levels
of navigational information processing, from physical measurements by sensors
(preliminary processing), through the combination of measurements from
different sensors (intermediate processing) to the estimation of position
coordinates and other navigational parameters (final processing). At each of
these levels, we use the same mathematical tools and the same computing
algorithm.
The
discrete Kalman filter, in a particular case, describes the system of two
equations [1], [2], [5], [6]:
- State equation (structural model)
- Measurement equation (measurement model)
where:
r
£
n, p £ m.
We
assume that the vectors of disturbances w
and v are Gaussian noise, with
normal distribution and a zero mean vector, and are mutually non-correlated. In
the case of colour noise (with a trend), the extended Kalman filter is applied,
where the disturbance trend is included as additional components of the state
vector.
The
equation of state describes the evolution of the dynamic system described in
the state space, while the model of measurements functionally combines
measurements with the system state. The solution to Equations (8) and (9),
taking into account the constraints imposed on the vectors of disturbances, is
the Kalman filter. Calculation of the state vector in the Kalman filter is
described by the following algorithm:
- Projection of the state vector:
where
- Covariance matrix of the projected state vector:
where Q is the covariance matrix of disturbances of the state (of vector w)
- Innovation process:
- Covariance matrix of the innovation process:
where R is the covariance matrix of
measurement disturbances (of vector v)
- Filter gain matrix (Kalman matrix):
- Estimated value of the state vector from filtering
after measurement
- Covariance matrix of the estimated state vector:
4. THE
STRUCTURE OF THE INTEGRATING FILTER
The
adopted mathematical model of ship movement and the configuration of
navigational devices
affect the structure of the Kalman filter algorithm. Let us assume, as in the
position estimation algorithm by the method of least squares, that position
coordinates are determined using GPS (parametric navigation), while
measurements in dead reckoning navigation are obtained from a gyroscopic
compass and Doppler log.
Let us
define the state vector as:
where:
The
transition matrix A of the
structural model will be:
where:
a - major semi-axis of
the Earth’s ellipsoid
e
- the first eccentricity of the Earth’s
ellipsoid
Another
element of the structural model is the covariance matrix of the state
disturbance vector Q. Its elements
define a priori distributions of disturbances of the estimated quantities. For
the state vector, as defined by Formula (10), the matrix of the state Q disturbances may assume this
form:
where:
The
quantities measured (measurement model) are the following parameters: position
coordinates from a GPS system
The
matrix of measurements is the Jacobi matrix, which has the following form:
The
matrix of measurement disturbance covariance (measurement vector) is also an
element of the measurement model. It is a band matrix because certain
quantities measured are not correlated with each other, e.g., GPS measurements
and components of speed from dead reckoning navigation, or gyroscope and log
measurements.
5. SUMMARY
The presented models and algorithms
illustrate two of many possibilities regarding the navigational application of
the method of least squares and the Kalman filter for the integration of
navigational data. In the Kalman filter, the state vector reproduces system
evolution (movement trend) on the basis of dead reckoning navigation. The main
advantages of the Kalman filter, in this case, are its recurrence, which is a
natural necessity in case of ship navigation, and the possibility of using the
ship movement data (its trend).
There are other approaches to Kalman
filtering, based on Monte Carlo simulations [4] and artificial intelligence
methods [10], which enable identification of the state model parameters and online measurements.
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Received 21.01.2017; accepted in revised form 13.04.2017
Scientific Journal of Silesian
University of Technology. Series Transport is licensed under a Creative
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