International Journal
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.
90
Fig. 1. Layout of two antennas GPS (DGPS)
We are searching the minimum of the following
function:
=∆∆∆∆ ),,,(
2121
yyxxf
min
2
1
2
2
2
1
2
2
→∆+∆+∆+∆ xyyx
, (1)
with constraint
[ ]
+∆+−∆+=∆∆∆∆
2
11222121
)()(),,,( xxxxyyxxg
[ ]
=−∆+−∆+
22
1122
)()( dyyyy
0, (2)
where:
∆x
1
, ∆x
1
– deviation (correction) from the
position of medium antenna A
1
,
∆x
2
, ∆x
2
– deviation (correction) from the
position of medium antenna A
2
,
d – distance between antennas ‘1’ and ‘2’
at the horizon plane.
In this case the Lagrange function will take the
following form:
+∆+∆+∆+∆=∆∆∆∆
2
1
2
2
2
1
2
22121
),,,,( xyyxyyxxL
λ
[ ]
+∆+−∆+
2
1122
)()({ xxxx
λ
[ ]
=−∆+−∆+ })()(
2
2
1122
dyyyy
+−+∆+∆+∆+∆=
2
12
2
1
2
2
2
1
2
2
)[( xxxyyx
λ
+−
+∆−∆+∆−∆−
2
1
2
2
121212
)()())((2 yyxxxxxx
])())((2
22
121212
dyyyyyy −∆−∆+∆−∆−
(3)
where:
λ
is Lagrange multiplier.
We introduce the following symbols to make the
notation simpler:
12
xxx −=∆
- the difference between mean positions
X-axis (4)
12
yyy −=∆
- the difference between mean positions
Y-axis (5)
222
12
2
12
)()( yxyyxxD ∆+∆=−+−=
- the distance between mean positions of the
antennas (6)
Now the equation (3) has simpler form:
=∆∆∆∆ ),,,,(
2121
λ
yyxxL
+∆+∆+∆+∆
2
1
2
2
2
1
2
2
xyyx
+∆−∆∆+∆ )(2[
12
2
xxxx
λ
+∆−∆∆+∆+∆−∆ )(2)(
12
22
12
yyyyxx
])(
22
12
dyy −∆−∆
(7)
Now the values of the following corrections will
be the solution to problem (1) with constraint (2)
x
D
dD
x ∆
−
=∆
2
1
,
y
D
dD
y ∆
−
=∆
2
1
, (8)
x
D
dD
x ∆
−
−=∆
2
2
,
y
D
dD
y ∆
−
−=∆
2
2
. (9)
The corrections calculated with functions (8) and
(9) are added to the mean position of the antennas
and in this way we obtain the corrected coordinates
of the antennas:
111
xxx ∆+=
′
,
111
' yyy ∆+=
, (10)
222
xxx ∆+=
′
,
222
' yyy ∆+=
. (11)
Formulas (10) and (11) define the corrected mean
positions of the antennas:
),(
111
yxA
′′′
and
),(
222
yxA
′′′
,
with constraint (2).
3 THE ANALYSIS OF LINEAR CONSTRAINTS
CASE
In the analyzed case of linear constraints of the mean
positions of the antennas, the following situations
can be taken into consideration:
1 the estimated earlier mean positions of the
antennas are not affected by relative systematic
errors, i.e. D = d and then according to formulas
(8) and (9) corrections of the coordinates will
equal zero;
2 the estimated earlier mean positions of the
antennas are affected by relative systematic
errors, i.e. D ≠ d and then one of the two cases
will be noted:
− D > d and the corrected mean positions of the
antennas are placed on a straight line joining
points A
1
, A
2
and within the segment
21
AA
,
91
− D < d and the corrected mean positions of the
antennas are placed on a straight line joining
points A1, A
2
and outside the segment
21
AA
.
Figure 2 illustrates the above described cases.
a) b)
c)
Fig. 2. The location of mean positions of the antennas in
relation to each other and the corrected mean positions of
the antennas
4 ESTIMATION WITH CONSTRAINTS OF THE
MEAN VALUES OF COORDINATES OF
GPS/DGPS POSITIONS
True measurements will illustrate our investigations.
The readings were taken on the antenna platform of
the Maritime University of Szczecin on 26 April
2006. The positions of the antennas were established
geodetically in ‘65’ system and then their coordinates
were transformed into WGS-84 system- points A
1
,
A
2,
whereas the measurements in the receiver were
taken in WGS-84 system only- A
1
(DGPS),
A
2
(DGPS). The positions of the antennas
)DGPS(
1
A
′
and
)DGPS(
2
A
′
were obtained once the corrections
of the coordinates resulting from the imposed
conditions on the distance between two antennas (in
this case d = 3.406 m) were taken into consideration.
Fig. 3. Estimation with constraints with the use of true
GPS/DGPS measurements
The deviations of the corrected positions from the
true positions of the antennas may result from the
errors of the systematic measurements of DGPS or
also (in this case) from the systematic error of
coordinates transformation from the ‘65’ system (the
catalogue of points of geodesic matrix) into WGS-84
system.
5 ESTIMATION WITH CONSTRAINS –
MOVING VESSELS
The method of estimation with constrains can also
be employed for dynamic measurements. So when
the measurements are not the estimated values (e.g.
from Kalman filter or from stochastic
approximation) then we will have to do with a
deterministic case – a conditioned extreme value.
This situation is illustrated by the pictures below
(Fig. 4. and Fig. 5.) The former presents combination
of two circulations put together. The outer curves
represent the original GPS measurements affected by
great errors and the inner curves represent
measurements which take into account constraints
(1.897 m). The enlarged fragment of circulation is
shown in Figure 5. Now, big systematic errors made
in GPS measurements can clearly be seen.
92
Fig. 4. Estimation with constraints with the use of true GPS
measurements - circulating vessel (‘eight’- shaped curve)
Fig. 5. The enlarged part of Figure 4- the outer curves represent
mean trajectories of given receivers and the inner curves
represent the trajectories of mean values with constraints
6 CONCLUSION
The presented method of estimation of the position
of ship’s coordinates with constraints can be used
when at least two navigational receiving systems are
employed. It can be spread to angular and linear-
angular constraints (three or more receivers). The
constraints can be defined either by direct linear
measurements or by angular-linear ones or can be
calculated analytically when the positions of
antennas in relation to ship’s construction elements
are known.
In case when the coordinates of ends of the
segment (two antennas) are measured and there is
only one linear constraint, then the solution is
represented by two points on one line with mean
positions of direct measurements and placed
symmetrically, in relation to them (Figures 2a-2c).
The shape is represented correctly and the direction
of the segment remains unchangeable. This means
that the direction of the segment is marked by
measuring error resulting not from one, but from two
receivers which with their positive correlation
(resulting from the constrains) may lead to the
increased accuracy of the defined direction/course
(true course) [1] and the spatial position of the
vessel. It has significant importance both in
hydrographic measurements and in navigation in
restricted areas.
REFERENCES
[1] Banachowicz A., Banachowicz G., Uriasz J., Wolski A.: A
Comparison of GPS/DGPS Relative Accuracy. 6th
International Conference “Transport Systems Telematics”.
Katowice-Ustroń, October 25-27, 2006.
[2] Baran L.W.: Teoretyczne podstawy opracowania wyników
pomiarów geodezyjnych. Wydawnictwo Naukowe PWN,
Warszawa 1999.
[3] Bertsekas D.P.: Constrained Optimization and Lagrange
Multiplier Methods. Academic Press, New York 1982.
[4] Bronsztejn I.N., Siemiendiajew K.A., Musiol G., Mühlig
H.: Nowoczesne kompendium matematyki. Wydawnictwo
Naukowe PWN, Warszawa 2004.
[5] Luenberger D.G.: Teoria optymalizacji. PWN, Warszawa
1974.
[6] Łoś A.: Rachunek wyrównawczy. T. I. PWN, Warszawa –
Kraków 1973.
[7] Nowak R.: Statystyka dla fizyków. Wydawnictwo Naukowe
PWN, Warszawa 2002.
