20 Scientific Journals 32(104) z. 2
Scientific Journals
Zeszyty Naukowe
Maritime University of Szczecin
Akademia Morska w Szczecinie
2012, 32(104) z. 2 pp. 20–23 2012, 32(104) z. 2 s. 20–23
The gradient method of determining vessel waterplane position
based on multiple laser sensor positioning system
Marek Duczkowski, Lucjan Gucma
Maritime University of Szczecin, Faculty of Navigation
70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: l.gucma@am.szczecin.pl
Key words: waterplane position, restricted area, gradient algotithm Abstract
Using laser rangefinders or other distance measuring tools offers an opportunity to create new ship waterplane position finding algorithms in restricted areas. The achieved by such system redundancy of positioning system will affect on ship’s safety and influence navigation which is dominated by one system (GPS and its derivatives). This paper is focused on developed optimization gradient algorithm and its improvement, algorithm with random start position.
Introduction
In marine transportation there is a strong de-mand to obtain position from as many sources as possible to provide navigation safety and redundan-cy of positioning system. With usage of laser range-finders or other distance measuring tools another methods of obtaining waterplane position can be developed.
Due to fact that vessel’s waterplane is irregular, traditional geometric methods cannot be used. Therefore, optimization algorithms should be con-sidered. Gradients methods gives good results when target function is known [1]. To construct target function waterplane has to be exactly described as set of points. Some criteria of choosing a number of points used in each iteration have to be given.
Developed algorithm was tested in several runs with changing starting conditions of ships waterplane position and direction. Method assumes that following information is available:
1) the geometry of ship waterplane as set of m- -points (xm, ym);
2) k-sensor position and its directions;
3) distances from o-laser sensors to ships water-plane;
4) approximate ships course (with up to 30 degrees error).
Waterplane determination problems due to geometry and sensor numbers
Sensors number influence
Position of waterplane can be obtained only from more than two laser sensors values. In figure 1 two situations when waterplane can not be deter-mined with use of two lasers, which have the same distance to waterplane, are presented.
Fig. 1. Two possible results when there are only two laser rangefinders
Adding more than two lasers will decrease num-ber of possible results, will cause better algorithm efficiency. Required minimum for this algorithm is three sensors number.
Sensors geometry influence
Obtaining waterplane position could be prob-lematic due to its irregularity and possibility of obtaining many possible results for given sensors
The gradient method of determining vessel waterplane position based on multiple laser sensor positioning system
Zeszyty Naukowe 32(104) z. 2 21
geometry. In most cases, when two or more aligned lasers finders are targeting from the same berth, the solving of equations and determination or water-plane is impossible. Example of this case is presented in figure 2.
Fig. 2. Two possible results when three laser rangefinders are in line
Gradient algorithm
Base gradient algorithm calculates in given iteration values of gradient, which coordinates are from definition for Cartesian coordinate system a partial derivative from target function in each changing direction [2]. In waterplane position searching algorithm directions are x and y positions of the waterplane and angle α (course of vessel):
α f , y f , x f f (1) Target function
Target function is a square of distances from measured distance points to a predicted waterplane point position, where the shifting vector values of vector P(px, py) and rotation angle α are variables,
for each laser rangefinder. In figure 3 point A is rotated around M point to point B and shifted by vector P to point C. G is a point descendant from distance measured by laser.
Fig. 3. Point prediction calculation method
Point A can be rotated around point M by using following formulas: y y y x x y x y y x x x M M A M A B M M A M A B ) cos( ) ( ) sin( ) ( ) sin( ) ( ) cos( ) ( (2)
Distance D(dx, dy) is calculated by:
y y y y y x x y x x x y y x x x G P M M A M A d G P M M A M A d ) cos( ) ( ) sin( ) ( ) sin( ) ( ) cos( ) ( (3)
Therefore, target function for optimization is described by:
n i xi yi d d f 1 2 2 (4)where: n – number of points measured by distance measuring tools.
Course of the vessel is assumed with 30° error and finally optimization function could be de-scribed as:
fmin when CC030 where:
C – ships actual course;
C0 – ships given (assumed) course. Gradient algorithm
In each iteration the nearest point on waterplane to measured distance point G is searched and this point is used as point A. Algorithm finish criteria is finding position with summary square distance lower than assumed or exceed assumed maximum number of loops. Algorithm is presented as block diagram in figure 4.
Fig. 4. Waterplane position searching gradient algorithm
Start Stop Input starting data Is criterium fulfilled
Find nearst point on waterline for each distance point
Shift all points on waterline by gradient multiplying by multiplying factor Calculate partial derivatives Calculate multiplying factor Yes No
Marek Duczkowski, Lucjan Gucma
22 Scientific Journals 32(104) z. 2
Multiplying factor value depends on previous gradient value or can be fixed. This factor is used to speed up algorithm and has little influence on algorithm results.
Gradient algorithm test
Gradient algorithm was tested for 5 points waterplane. There were 3 sensors in configuration presented in figure 5.
Fig. 5. Point prediction calculation method
In test 1000 randomly selected ship’s position was generated and optimization process conducted. The algorithm enables to determine correct results of only 47% of tested cases.
Gradient algorithm with random start position
The main reason of errors in gradient algorithm was related to distance of the waterplane to which the laser sensors established the position. After analyzing first test, it was observed that changing start position in many cases greatly improve algo-rithm effectiveness. Therefore, gradient algoalgo-rithm was improved by new block: random generation of start position within given boundaries (Fig. 6) where Dmax is a distance between waterplane zero point and furthest waterplane point.
Fig. 6. Random starting position region
Gradient algorithm with random start position calculation
Random position is generated in loop. In each loop base gradient algorithm is started. Gradient algorithm with random start position has the same finish criteria as base gradient algorithm, but there is also a maximum number of random position cri-terion. Algorithm is presented in figure 7.
Fig. 7. Waterplane position searching gradient algorithm with random start position
Gradient algorithm with random start position test Improved algorithm was tested using the same method as previous algorithm. Efficiency increased to 71% of all cases. Although algorithm efficiency was increased by 24%, in some cases algorithm can block itself in certain position. This behavior is caused by situation when the nearest edge on waterplain to position given by sensor is the proper one. Therefore, in some situation it cannot be changed, because also in further iterations edges used in calculation are the same. This behavior
Dmax G2 G3 G1 Dmax Dmax Dmax y x Start Stop Input starting data Is criterium fulfilled No
Find nearest point on waterline for each distance point
Shift all points on waterline by gradient multiplying by multiplying factor Calculate partial derivatives Calculate multiplying factor Randomizing start position Is criterium fulfilled No Yes Yes
The gradient method of determining vessel waterplane position based on multiple laser sensor positioning system
Zeszyty Naukowe 32(104) z. 2 23
exists only in some cases. By increasing the number of waterplane points, algorithm differences between edges will be smaller and algorithm will change edges used in calculation more easily.
After increasing the number of points in waterplane to 13 effectiveness of position determi-nation increased to 76%.
Fig. 8. Waterplane positions created by gradient algorithm
In figure 8 are presented two sample vessel posi-tion finding algorithm run with randomly generated start position and starting assumed course. Random
positions given by sensors and initial data in this particular calculations are presented in table 1.
Table 1. Position given by sensors
Test X position Starting Y position Starting Assumed course Sensor No. X Y
1 198 211 30 1 52 150 2 105 174 3 41 98 2 42 11 248 1 21 11 2 67 54 3 84 2 Conclusions
In this paper gradient optimization method was applied to determine of ships waterplane. Such method could be the part of novel positioning sys-tem consists of several rangefinders located on-shore. System could be used in restricted water areas. One of the main advantage of presented posi-tioning system is independency from GPS based systems. Presented algorithm gives promising ef-fectiveness in waterplane position determination (around 80% in given sensor configuration and low-level detail waterplane). Further improvement and research could ameliorate algorithm execution efficiency and effectiveness and application togeth-er with low cost compasses information which re-duces uncertainty of ships course.
Searching position of waterplane always will be connected with some uncertainty due to fact that in many cases there are more than one possible result, so global optimization algorithms, like for example GA, could be useful in further studies.
References
1. FLETCHER R.: Practical Methods of Optimization. John
Wiley and Sons, 1987.
2. ELIJAH POLAK: Optimization: Algorithms and Consistent
