on Marine Navigation
and Safety of Sea Transportation
Volume 1
Number1
March 2007
89
The use of estimation of position coordinates
with constraints in navigation
A. Banachowicz
Gdynia Maritime University, Gdynia, Poland
G. Banachowicz
Maritime University of Szczecin, Szczecin, Poland
ABSTRACT: This article presents method of estimation of position coordinates with the use of conditions
imposed on measurements. The application of two (or more) systems used to define position and taking into
account the way they are related, makes it possible to improve both accuracy and reliability of the measured
navigational parameters. The method of estimation with constraints is based on the method of determining
extremes with constraints of many variables. In geodesy, this method is known as adjustment by conditions.
1 INTRODUCTION
In classical navigation the position of a vessel is
defined as one chosen point. Most often it is the
position of an antenna of the radio-navigational
system or the measurements are treated as one
common point, e.g. the centre of mass. Nowadays
when the navigating techniques used for establishing
ship’s position are so easily accessible and relatively
cheap, more than one device can be employed at the
same time. In order to work out such measurements,
a method which makes use not only the measured
data but also mutual location of the antennas can be
employed. The antennas can be positioned in a
different configuration in relation to one another, i.e.
they can be linear, angular or angular and linear.
That is why in such a case estimation with
constraints, known as estimation with constrained
equations [7] or, in geodesy, as adjustment by
conditions [2], [6] can be used. This method has its
deterministic origin in constrained optimization
which makes use of Lagrange multipliers [3], [4],
and [5]. There are the following advantages in this
method: fewer errors of systematic measurements of
ship’s coordinates and the increased accuracy and
reliability of the calculated ship’s trajectory.
This article presents method of estimation of
position coordinates with the use of constraints
imposed on measurements using true measurements
of navigational satellite systems GPS (DGPS). The
research has been carried out in static conditions, on
an antenna platform located at Maritime University
of Szczecin at places with known coordinates and
mutual positions.
2 ESTIMATION WITH CONSTRAINTS
The core of the method of estimation with
constraints is to find the minimum of a given
function where the variables of that function are
constrained in a defined way. In our case the sum of
squares of deviations (the rule of the smallest
squares) from the position of two average receivers
antennas with known distance between them was
minimized. This situation is illustrated by Figure1.
Antenna ‘1’ has the mean position A
1
(x
1
, y
1
) and
antenna ‘2’ – A
2
(x
2
, y
2
). Additionally, we know that
the true distance (binds) equals d.