The Mathematics of Collision Avoidance in Two Dimensions

219

107

Efficient marine navigation through obstructions is still one of the many problems faced by the mariner. Many accidents can be traced to human error, recently increased traffic densities and the average cruise speed of ships impedes the collision avoidance decision making process further in the sense that decisions have to be made in reduced time. It seems logical that the decision making process be computerised and automated as a step forward to reduce the risk of collision. This article reviews the development of collision avoidance techniques and path planning for ships, particularly when engaged in close range encounters. In addition, previously published works have been categorised and their shortcomings highlighted in order to identify the ' state of the art ' and issues in close range marine navigation. K E Y

Section: Studies In Collision Avoidancementioning

confidence: 99%

Review of Collision Avoidance and Path Planning Methods for Ships in Close Range Encounters

Tam

Bucknall

Greig³

2009

219

107

“…Meanwhile, S. H. Hollingdale, a mathematician and colleague of Calvert, had published a paper 3 showing that Calvert's system of manoeuvres was mathematically consistent and, further, that it was the only system which met the requirement that manoeuvres should depend only on the bearing of a threat.…”

Section: John Kemp Writesmentioning

confidence: 99%

Manœuvres to Ensure the Avoidance of Collision

Calvert

1997

This paper was first published in 1960 (Vol. 13, p. 127). It is followed by comments from John Kemp. The paper has been abridged, including the omission of section 5 which described a proposal for a new radar display.When the problem of collision in the air is discussed, it is usual to start by pointing out the enormous closing speed of two modern aircraft meeting head-on, and to conclude from this that avoidance on the ‘see and be seen’ principle has ceased to be possible. The fact is, however, that the great majority of mid-air collisions (about 85 percent) occur within five miles of an airport and the typical case is not the head-on one, but the case in which the two aircraft crab into one another from a direction which may be anywhere around the whole enclosing sphere. Since the field of view of the aircrew covers only about 20 percent of the enclosing sphere, the aircrew of colliding aircraft seldom see each other. It would seem, therefore, that the ‘see and be seen’ principle never did afford much protection, even when speeds were low. In other words, the fact that the number of mid-air collisions in Europe has hitherto been small is not primarily due to seeing and evading, although this sometimes happens, but to the fact that the airspace is very large compared to the volume of all the aircraft in it at any given time. However, as traffic densities go up, the risk rapidly increases, and in congested airspace, such as that around New York, the problem of avoiding collision has already become acute. In the period 1948–57, there were 159 mid-air collisions in the United States, and many of these made headlines in the world press. One can imagine the public outcry if two large transports were to collide over a housing estate; but unless something effective is done, something like this will presumably happen eventually. At very high altitudes the ‘see and be seen’ principle certainly fails, by day, because the speed will be high, and in addition, the range at which a pilot can see an object the size of an aircraft may be less than 1½ miles due to what is sometimes called ‘high-altitude myopia’.

“…Formula (3) shows that the uncertainty in the true value of the miss distance, for any size of bearing error 77, steadily decreases as the observations are continued down to shorter and shorter range. We shall take this latter value (3 standard deviations) as giving 'practical certainty' (any other value may, of course, be selected if preferred).…”

Section: H Rmentioning

confidence: 99%

“…3) as the angle through which the relative velocity vector rotates when one or both ships manoeuvre. To extend the results to a near-miss situation, one has only to redefine the angle a (see p. 24s of ref.…”

mentioning

confidence: 99%

The Effect of Observational Errors on the Avoidance of Collision at Sea

Hollingdale¹

1964

Self Cite

At a meeting of the Technical Committee of the Institute held on 9 January 1963 it was suggested that recent theoretical treatments of the collision problem could usefully be extended to include a discussion of the near-miss situation and the effect of observational errors. The basic mathematical relations for near-miss encounters have been set out in this Journal on several occasions, notably by Sadler and Morrell, and in graphical form by Wylie. The recent paper by Parker deals with the effects of both systematic and random errors of radar observations of relative range and bearing.My previous discussion of the collision problem was presented in terms of the idealized situation where two ships are on an actual collision course, in which case the sight line is in the same direction as the relative velocity vector (the relative track). To extend the results to a near-miss situation, one has only to redefine the angle α (see p. 24s of ref. 3) as the angle through which the relative velocity vector rotates when one or both ships manœuvre. I have pointed this out on a previous occasion: ‘The two craft need not be on a collision course; this definition of α applies equally well to a “miss” situation provided that no reference is made to the sight line.