A FIELD OF VELOCITY VECTOR IN CONTROL
AREA OF ENTRY INTO THE INTERSECTION
Kasyk L.
Institute of Mathematics, Maritime University of Szczecin, Szczecin, Poland
Abstract: Using the concept of a ship’s domain at a waterway intersection, the ships’ traffic
equations were determined, describing a situation where they pass each other at the intersection without collision. A field of limit velocity vectors was established, at which the vessel can pass through the intersection without collision. On the basis of designated vector fields, the control area of entry of the subordinated vessel into the intersection was defined.
1. Introduction
The present paper concerns the intersection, which is a special place in the waterway system [1, 3, 9, 10]. Each passage of the ship through the intersection is bound with a certain risk [2, 4], which is due to the fact that on every way of one unit, another can be encountered. Out of two vessels approaching each other on crossing routes, always one of them is privileged in relation to the other (it has priority). This can result from general rules of moving on sea routes, or from regulations controlling traffic on a given water area. Hence, the subordinated unit must proceed in such a way as to let the privileged one pass. When the units are close to the intersection, the subordinated vessel has to perform a manoeuvre, in order to avoid a collision situation (excessive approach) with the unit proceeding crosswise. It is therefore very important to determine such speed (with relation both to value and direction), which would enable the ship at a definite point of the intersection to safely pass the unit sailing crosswise.
The area, where the vessel must find itself before performing the manoeuvre, in order to have the optimal conditions for starting it, is called the control area of entry [5, 7, 8, 12]. In the case of passing the privileged unit, the matter is to take such a position in the area before the intersection that the subordinated unit, proceeding with definite parameters, avoids a collision situation (or excessive approach) with the unit sailing crosswise. In this paper, the control area of entry is understood as a set of points in the area around the
intersection, at which the vessel has the speed (value and direction) enabling it to safely pass the privileged unit.
1. A diagram of the intersection
Waterways are mostly of irregular shape, but for the purpose of modelling, the following intersection diagram has been assumed (Fig. 1).
B A
P
Fig. 1. An intersection diagram
Two fairways have been marked on the diagram, crossing each other under the angle . Point P is the place where the axes of the fairways intersect, where the vessels move. In the situation presented in Fig. 1, vessel A must give way to vessel B, if it is close to the intersection. If, on the other hand, A’s navigator sights vessel B far from the intersection (this can happen by visual observation or by radar), he has to estimate if he manages to pass ahead of B’s bow in the prescribed distance or not. The vessel’s domain at the intersection can serve this purpose [6, 9, 11]. This is the probable area of the ship’s being at the intersection, determined from information on a given traffic regulation scheme, at the moment of assessing the safety of the ships passing each other at the intersection. In addition, a system of coordinates was constructed on the intersection diagram, which will make it possible to establish the velocities, depending on the point’s coordinates at the intersection. x O2 y P O1
The center of this system is the point where the fairway axes intersect. Axis O1 is the axis of abscissae, and the line perpendicular to the axis of abscissae crossing point P is the axis of ordinates (Fig.2).
2. Limit velocities
It has been assumed in the paper presented that the privileged vessel B proceeds in the fairway axis with constant velocity vB; vessel A, on the other hand, also has a constant
velocity which creates angle with axis O1, which can assume positive or negative values in accordance with the principles of analytic geometry (Fig.3).
x O2 y P O1 vA vB
Fig. 3. Diagram of vessel velocities at an intersection
The concept of ship’s domain and danger zones created by the author [6, 9, 10, 11] was used to determine vessel traffic equations, which describe the situation of vessels passing through an intersection without collision. The limit situation of the domains’ contingence, presented in Fig. 4, signifies that in the anticipated time of reaching the intersection, unit A, which is in domain DA, will pass behind unit B’s stern, which is in domain DB.
DA
DB
Fig.4. The moment of highest proximity of domains at the crossing of the ferry behind the fairway unit
In the case of = 0, the analysis of the limit situation presented in works [8] and [11] was used as basis. The situation presented in Fig. 4 will occur, when vessel A proceeds at limit velocity vP1 [8]. This is the speed at which unit A, being at any point (x, y) of the
intersection water area will pass astern the fairway unit. In order for vessel A to pass safely through the intersection astern of unit B, sighted at distance dB from the
intersection, it must proceed at a speed not higher than vA1:
1 2 2 2 ctg 2 cosec 2 2 A CA A B B B CB l L x y v v y d l L (1) where: vB – speed of unit B;
dB – distance from the intersection at which unit B was sighted;
LCB – unit B’s LOA;
LCA – unit A’s LOA;
lB – distance between unit B ellipsis centre and point P, at the moment of the
domains’ contingence;
lA – distance between unit A ellipsis centre and point P, at the moment of the
domains’ contingence.
In the case of unit B being still far from the intersection, vessel A’s master will be able to decide to pass ahead of the privileged unit’s bow. In order for vessel A, which is at point (x, y) to pass safely ahead of unit B’s bow, the value of its speed must not be smaller than limit velocity vA2, determined by equation [8]:
2 2 2 2 ctg 2 2 2 cosec A CA A B B B CB l L x y v v d l L y (2)
In the case of 0, it can be assumed that the real angle between vessel A’s velocity and axis O2 is equal to . Hence, the limit velocities for vessel A, which is at point (x, y) and proceeding in direction 0 are equal to:
1 2 2 2 ctg 2 cosec 2 2 A CA A B B B CB l L x y v v y d l L (3) and
2 2 2 2 ctg 2 2 2 cosec A CA A B B B CB l L x y v v d l L y (4)3. Vector fields of velocities
Equation (3) determines the vector field of the subordinated unit’s limit velocities, passing astern the privileged unit. In this case, velocity is determined by the limit value vA1 and
Equation (4) determines the vector field of the subordinated unit’s limit velocities, passing ahead the bow of the privileged unit. In this case, velocity is determined by the limit value vA2 and angle of inclination to axis O1. For the established value of angle , each point at the intersection with coordinates (x, y) can be assigned two limit velocities. Thus, we obtain two vector fields of limit velocities, which enable unit A to pass through the intersection without collision (on a suitable level of confidence).
B
Fig. 5. Vector field of vessel A’s limit velocity values when passing astern unit B
By means of the vector fields determined above, the subordinated unit, approaching the intersection at speed vA is able to assess the safety of passing through the intersection. If
vessel A, being at a point (x, y), proceeds at speed vAvA1 or vAvA2, this velocity safeguards a safe passage through the intersection (on a suitable level of confidence). On the other hand, when vA1 <vAvA2, passing through the intersection without altering the traffic parameters is very likely to bring about a collision situation with a unit proceeding crosswise.
Equation (3) can also serve the determination of angle , with established value of vessel A’s velocity. Replacing in formula (3) limit velocity vA1 with velocity vA of unit A, and
angle with critical angle 1, the following formula was obtained at first:
1 1 2 cosec 2 2 2 2 2 ctg A B B CB B A CA v y d l L v l L x y (5)Determining angle 1 from equation (5) is rather troublesome. Therefore, the following substitutions were applied for simplicity’s sake:
2 2 2 2 2 2 A A B A B A CB B A B CA B B v y a v d v l v L v l v L v x b v y c (6) After applying these substitutions, formula (5) can be reduced to the following shape:
1
11
cos sin sin a b c (7)Hence, angle 1 is expressed by the formula:
4 2 2 2 2 1 arcsin 2 2 ab c b c a c b c (8)
With established value of velocity vA the situation of domains’ contingence (Fig.4) will
occur for the value of angle 1 determined in equation (8). This means that the passage of vessel A will only take place astern vessel B, when the course of vessel A creates an angle with axis O1 smaller than 1.
Similarly, transforming formula (4), the value of critical angle 2 was obtained:
4 2 2 2 2 2 arcsin 2 2 ae c e c a c e c (9) where 2 2 2 2 2 2 A A B A B A CB B A B CA B B v y a v d v l v L v l v L v x e v y c (10)
For the established value of velocity vA, each point at the intersection with coordinates (x,
y) can be assigned two critical angles. Thus, two vector fields of limit velocity directions are obtained, which make it possible for unit A pass through the intersection without collision (on a suitable level of confidence).
B
With established velocity value of vessel vA, the master of the subordinated unit
approaching the intersection is able to assess the safety of passing through the intersection. If vessel A, being at a point (x, y), proceeds with velocity vA in a direction
which forms an angle with axis O1 smaller than 1 or is larger than 2, then this direction
safeguards (on a suitable level of confidence) a safe passage through the intersection. On the other hand, if 1 < 2, passing through the intersection without altering the traffic parameters is very likely to bring about a collision situation with a unit proceeding crosswise.
4. Control area of a ship’s entry into an intersection
The vector fields determined above can serve the determination of control entry area into an intersection of a subordinated vessel. With established angle 0 between velocity of vessel A and axis O1 and for a specific speed v0, all points, for which
vA1v0 or vA2v0
and
1 0 or 20
, for a control entry area of a vessel into the intersection (Fig.7). B A P l k m n
Fig. 7. Control entry area into an intersection of a subordinated vessel
The control area of a subordinated vessel’s entry into an intersection is presented in Fig. 7 in the form of darker polygons. Straight lines k and l was determined from equations (3) and (4). Straight m is going by point (–1/2BActg , 1/2BA) in direction 0, and straight n is going by point (1/2BActg , –1/2BA) in direction 0, where BA is a width of fairway of unit
A. This area is limited with straight lines k, l, m, and n with the equations:
0
0
0
0 0 0 0 0 0 2 2 2 : cosec ctg 2 cosec 2 ctg B B CB A B B CA B B B d v l v L v l v v L v k y x v v v v (11)
0
0
0
0 0 0 0 0 0 2 2 2 : cosec ctg 2 cosec 2 ctg B B CB A B B CA B B B d v l v L v l v v L v l y x v v v v (12)0 0 1 1 : tg tg tg 2 A 2 A m y x B B (13) 0 0 1 1 : tg tg tg 2 A 2 A n y x B B (14)
5. Conclusions
The velocities determined in the paper can be a useful tool of an anti-collision system for vessels passing through an intersection. They can also contribute to reducing the operational costs for vessels passing through an intersection (no stops in front of the intersection to let a privileged unit through, adapting speed to the situation at the intersection).
For offices taking care of waterways, Control Areas of Entry can serve the purpose of assessing the importance of particular fairway sections with respect to safety of navigation; in consequence they can indicate water areas that require larger outlays for their maintenance or modernization; with restricted means, this will permit a more rational use of accessible funds.
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