Zeszyty Naukowe 32(104) z. 2 137
Scientific Journals
Zeszyty Naukowe
Maritime University of Szczecin
Akademia Morska w Szczecinie
2012, 32(104) z. 2 pp. 137–140 2012, 32(104) z. 2 s. 137–140
Making use of floating marks for position determination
Krzysztof Naus, Arkadiusz Narloch
Polish Naval Academy, Institut of Navigation and Hydrography
81-103 Gdynia, ul. Śmidowicza 69, e-mail: {Krzysztof.Naus; Arkadiusz.Narloch}@amw.gdynia.pl
Key words: free adjustment method, floating marks, position accuracy Abstract
Evaluation of accuracy of position coordinates’ determination in respect to floating marks, with free adjustment method applied, has been a target of the paper. General theory of geodesic adjustment – classical one based on least squares method and free adjustment – is presented in its first part. Main part contains description of the research. It was conducted based on geometric navigational structure which elements had been formed by beacons placed in the Bay of Gdańsk. It consisted in simulating results of measurements made in respect to marks of this structure (bearings and distances) and calculating the position coordinates based on them, as well as its accuracy. The calculations were made with the method of classical geodesic adjustment and free adjustment applied. Discussion about obtained results of the research and general conclusions are given in the ending part of the article.
Introduction
When determining watched position of a ship, one usually assumes that beacons based on which the position is to be determined are permanent points of known locations. However, it often happens in nautical practice that there are only floating navigational marks (e.g. setting out fairways, systems of traffic delimitation) available there to conduct the navigation. So far, no attempts of determining the ship position based on the floating marks have been undertaken, supposing that they may change own positions as a result waving, wind or sea currents. To generalize, one may adopt a thesis that coordinates of the floating marks are burdened with errors characterizing random displacement vectors.
Hence, application of position determination methods, in which displacement of marks making the so-called geometric measuring structure is assumed, is an important area of concern.
Method of the research
Method of geodesic adjustment should be applied in order to determine an optimal ship position from supernumerary results of measure-ments. Classical adjustment takes place only in
situation when the geometric structure corresponding to the observational system under the adjustment does not contain any degrees of freedom (the marks don’t change their positions).
In accordance with definitions given in [1], one may distinguish external degrees of freedom (SWU), describing possible displacements of the entire geometrical structure in respect to the coor-dinates’ system axis and its turn, as well as internal degrees of freedom (SWW), resulting from possible reciprocal displacement of elements making this structure. Overall number of the freedom degrees is expressed as SW = SWU + SWW. Should the beacons be considered permanent points, the geo-metrical structure assigned to the observational system does not contain the degrees of freedom.
If one assumes that navigational marks adopted for the measurements are floating objects, so it may not change its position, then the classical method of adjustment should be replaced with the free adjustment method. In this case, coordinates of the floating marks are considered only approximate values and the geometrical structure becomes a free structure. Not only increments up to the approxi-mate coordinates of a ship but also increments up to the known, previously considered permanent, coordinates of the beacons, are unknown data.
Krzysztof Naus, Arkadiusz Narloch
138 Scientific Journals 32(104) z. 2
The adjusting problem receives the following form in the theory of free adjustment:
0 for function objective additional min. ˆ ˆ problem classical . min ˆ ˆ ˆ L ˆ 1 2 0 2 0 2 1 2 1 d m m X T X T X X X ob ob d P d V P V P Q C d d A A L d A V X x x (1) where:A – matrix of coefficients of corrections’
equations; X
dˆ – estimator of real values of quantities
under measurement;
L – matrix of free terms of corrections’
equation;
m02 – estimator of variance coefficient;
P – matrix of weights of measurement
results;
PX– matrix of weights of points’ coordinates.
Following up the problem (1) solution, after de-termining such an estimator of increments dˆ that, X
V P
VˆT ˆ = min., next optimization problem, regu-lated with a matrix of weights of all points’ coordi-nates, may be formulated:
X T X X T X X X X X d P d d P d d Δ d B X X ˆ ˆ ) ( min 0 ˆ d (2) where:
A1PA1,A1PA2 B T T (3) PL AT 1 Δ (4)Equation (2) is an optimization problem of conditional method (correlative). To solve the above problem, it is necessary to replace the objec-tive function T X
XP d
dˆ Xˆ with a secondary function of a form of: ) ˆ ( 2 ) ˆ ˆ ( ) ˆ (d d PXd κT BdX Δ X T X X (5) where:
κ – vector correlative of Lagrange multipliers. This way, the secondary optimization problem occurs: min ) ˆ ( 2 ) ˆ ˆ ( ) ˆ ( , 0 ) ˆ ( Δ d B κ d P d d Δ d B X X T X T X X X (6)
After elementary conversions of the above dependencies, it results with:
Δ B BP B P dˆ X1 T( X1 T)1 X (7) where: T B BP Ξ X1 (8)
Moreover, assuming that matrix of weights of coordinates of all points under adjustment is an identity matrix: 2 1 0 0 X X X P P P (9)
Developing the expression (7), it results with:
ΞA PL PA A PA A P P d d d X X T T T X X X 1 1 2 1 1 2 1 2 1 0 0 ˆ ˆ ˆ (10)
Estimator of variance coefficient: d r n m T Vˆ PVˆ 2 0 (11)
In order to perform analysis of accuracy, cover-ing among the others: determination of mean errors of adjusted coordinates, errors of position or confi-dence ellipses of the network’s all points, it is necessary to determine the estimator’s covariance matrix Xˆ X0dˆX, so the covariance matrix of
coordinates of all points of the free geometrical measuring structure.
Determination of covariance matrix with the vector’s faultiness not taken into consideration X0:
1 1 1 1 1 1 2 0 ) ( ˆ ) ( ˆ ˆ ˆ X X d X C P B Ξ A PAΞ BP C T T bb bb X m (12)
where: CˆXˆ(bb) – covariance matrix of adjusted coordinates (with faultiness of approximate coordi-nates not taken into consideration).
Determination of covariance matrix with the vector’s faultiness taken into consideration X0:
ˆ ( ) 1 1 ) ( ˆ ˆ ˆ ˆ 0 r T bb zb X X dX X C I B Ξ BP C C (13)where: CˆXˆ(zb) – covariance matrix of adjusted coordinates, with faultiness of approximate coordi-nates taken into consideration).
The following dependence is applicable for cal-culation of mean error of the ship adjusted position:
2 ˆ 2 ˆ S S Y X poz m m m (14)
Making use of floating marks for position determination Zeszyty Naukowe 32(104) z. 2 139 where: S X mˆ , S Y
mˆ – stand for mean errors of
adjusted coordinates (Xˆs, Yˆs) of the ship position. Values S X mˆ , S Y
mˆ are obtained by determining estimator of covariance matrix from dependencies (12) and (13).
Elements of the confidence ellipse, for a real position of the point P, are determined with a use of the equations:
F
m
a
1 1 02
, b m 1F 2 0 2 (15) where:a – big semi-axis of the ellipse; b – small semi-axis of the ellipse;
PX P 2 1 1 (16)
PX P 2 1 2 (17)
2 4 2 XY X P P P (18)Torsional angle of the ellipse:
P P P X XY arctg 2 2 1 (19) Whereas: PA AT Y XY XY X P P P P (20) Conducted research
The tests were taken in the Bay of Gdańsk. The ship „P” was going along a traffic delimitation system, heading for the port of Gdynia. Buoys: GN (S1), Bingstead (S2), HL – S (S3) (Fig. 1) made the
geometrical measuring structure.
Fig. 1. Layout of marks S1, S2, S3 in respect to the ship P
The test consisted in simulating results of mea-surements done onto the marks of the geometrical measuring structure (bearings and distances) and, based on them – in calculating coordinates of the ship position’s coordinates and its accuracy. The calculations were made with a method of classical and free adjustments applied.
The following coordinates of the marks and ship were adopted for the calculations (Tab. 1).
Table 1. Coordinates of positions of marks and ship Name of point Geographical coordinates (ellipsoid WGS 84) Coordinates in UTM system [] [] X [m] Y [m] S1 – Buoy GN 54 31.9’ N 018 47.9’ E 6044945.22 357517.97 S2 – Gdynia Bingstead Buoy 54 33.2’ N 018 42.3’ E 6047473.69 351577.63 S3 – Buoy HL – S 54 35.3’ N 018 47.9’ E 6051174.88 357713.13 Ship 54 34.3’ N 018 50.8’ E 6049296.85 360824.34
It was assumed that the onboard navigational equipment determined (from the position P) bearing and distances to the marks: S1, S2, S3 (Tab. 2). Table 2. Measured values of bearings and distances
In addition, the following values were adopted for the calculations:
– mean error of the bearing’s measurement N was 0.2° (e.g. for measurement done with gyro-compass);
– mean error of the distance measurements d was 50 metres (e.g. for measurement done with radar);
– mean error of floating marks’ positions XY was 25 metres (value of the error was determined with a use of data about the marks’ positions gathered in the AIS system for a period of 6 months).
Based on the measured values of the bearings and distances, adopted, approximate ship position and adopted measurement errors, as well as errors of positions of marks of the geometrical measuring structure, coordinates of the position were calcu-lated – with a method of classical and free
adjust-Mark Real bearing [] Radar distance [m]
S1 215.5 5370 S2 257.0 9445 S3 300.0 3722 1 S 2 S 3 S P
Krzysztof Naus, Arkadiusz Narloch
140 Scientific Journals 32(104) z. 2
ments. Also mean errors and elements of the confi-dence ellipses for both positions obtained after the adjustments were determined. Computer pro-gramme named “Mathcad”, in which mathematical dependencies assigned for both methods of the position adjustment and evaluation of its accuracy were implemented, was used for the calculations (see section “Method of the research”).
Results of the calculations are given in tables 3 and 4.
Table 3. Coordinates of position after adjustment
Method of adjustment Coordinates in the UTM system
X [m] Y [m]
Classical adjustment 6049214.00 360851.60 Free adjustment 6049179.66 360837.64 Table 4. Accuracy parameters of position after adjustment
Method of adjustment Mean error of the position
Elements of the confidence ellipse (for = 95)
“a” – big
semi-axis “b” – small semi-axis torsional angle of the ellipse
[m] [m] [m] [°] Classical adjustment 45.9 152.5 77.5 –5.5 Free adjustment 65.3 757.5 385.7 –5.5 Results
Authors are aware of the fact that position should not be determined based on floating naviga-tional marks. However, it should be noted that the floating navigational marks are monitored by various navigational systems and the information is transferred on board of a ship, for instance, via the AIS system. Therefore, ways of usage of the floating marks in navigation should be verified, including their use in determining positions in respect to the marks.
Obtained results, i.e. value of the position’s mean error after free adjustment and size of its
confidence ellipse are somewhat bigger than the results obtained for position determined from the same measurements but executed in respect to permanents marks.
Accuracy of the position determination from floating marks may be improved by decreasing a number of freedom degrees adopted for the adjustment. Additional knowledge about vector of displacement of the mark’s position may enable this. It may be obtained, for example, from the drift’s and current’s parameters measured / calcu-lated on board of a ship or from up-to-date informa-tion about posiinforma-tions of the marks taken from the AIS system.
IMO requirements in respect to the position’s mean error state that the observed positions of ships representing draughts bigger than 15 metres should be determined with an error not bigger than 0.5 nautical mile, i.e. 926 m – for the entire coastal zone of Baltic Sea’s water region. Therefore, one may take a chance in affirming that the proposed mathematical apparatus can be used in determining position from the floating marks after meeting the IMO requirements regarding accuracy [4].
References
1. WIŚNIEWSKI Z.: Methods of elaborating measurements’ results in navigation and hydrography. Polish Naval Academy, Gdynia 2004, 279–280.
2. SZUBRYCHT T.,WIŚNIEWSKI Z.: Identification and correc-tions of beacons’ coordinates burdened with gross errors of placement. Scientific Journal No. 1 of the Polish Naval Academy, Gdynia 2004.
3. WIŚNIEWSKI Z.: Matrix algebra. Probabilistic and statistic basics of surveying calculations and data adjustment. ART, Olsztyn 1999, 7–44, 123–130.
4. GÓRSKI S., JACKOWSKI K., URBAŃSKI J.: Evaluation of accuracy of navigation performance. Maritime University, Gdynia 1990.
5. KOPACZ Z.,MORGAŚ W.,URBAŃSKI J.: Evaluation of ship’s position accuracy. Polish Naval Academy, Gdynia 2007, 78–80.
