The Mathematics of Collision Avoidance at Sea

Abstract

i. INTRODUCTION. Of the many factors affecting the collision problem, the mathematics of relative motion is the one most susceptible to formal analysis. Moreover, this 'collision geometry' must form the basis of any investigation into the instrumental, operational and human factors involved. It is therefore surprising that no complete description of the (mathematically) simple relationships is readily available. The object of this paper is to present such a connected account of the mathematics of relative motion, in so far as it appears relevant to the collision problem and within the limitations of time and space available. N The mathematics does nothing more than express the consequences of certain specific data in a form suitable for examination, thus simplifying the formulation of guiding rules or of particular courses of action. Complications arise solely through the large number of variables involved and the many combinations of circumstances for which analysis is required.The analysis is restricted to the relative motion of two ships initially moving with constant speeds on near-collision courses, as affected by changes of course by either ship, or by each simultaneously. One or two illustrations are given of the kind of deductions that might be made from this analysis. It must, however, be emphasized that the deductions are derived solely from a consideration of the geometry of the problem; they take no account of the many practical factors involved and completely ignore the Collision Regulations.But it must equally be stressed that a knowledge of the purely academic geometrical consequences, as here given, is an essential factor in the complete study.2. RELATIVE MOTION. The study is here restricted to the relative motion of two ships moving with constant velocities on near-collision courses. The notation, and sense of measurement of the angles, are as shown in Fig. i. For this purpose only relative bearings and courses are needed, and they are all referred to the course of the first ship. The second ship is always regarded as being to starboard, but it is clear that all necessary data can be derived for all cases. • : O, P are the instantaneous positions of the two ships A, B, moving at speeds, u, v on courses inclined at an angle C. The range and bearing of P from O are r and 0. The motion of B relative to A is along PD, bearing i8o° + S, with speed w. The inclination of the relative course PD to PO is S, which is necessarily a small angle if a collision is likely. D is the point of closest approach; the distance of closest approach OD is d, and 306'

Sensitivity of Safe Trajectory in a Game Environment on Inaccuracy of Radar Data in Autonomous Navigation

“…The basis of the positional game involves assigning the maneuver strategy of the own object to the current positions of p ( t k ) objects in the current step k . As such, the process model considers all possible changes in the course and speed of the encountered objects during the control [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42] (Figure 2).…”

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

“…To give an idea of the numerical magnitudes of p and rj, we may quote Sadler's figures, 5 namely p = o-o3 (a 3 per cent error), and •q =0-03 (a 2° bearing error, four times as much as that assumed by Parker" or WylieS). Whatever may be their actual size, if p and 17 are of the same order of magnitude, formula (1) shows that, provided mjry and m\r 2 are fairly small (say, less than i/y), range errors make a negligible contribution compared with that made by bearing errors.…”

“…At a meeting of the Technical Committee of the Institute 1 held on 9 January 1963 it was suggested that recent theoretical treatments of the collision problem 2 . The basic mathematical relations for near-miss encounters have been set out in this Journal on several occasions, notably by Sadler 5 and Morrell,6.7 and in graphical form by Wylie.8 The recent paper by Parker* 1 deals with the effects of both systematic and random errors of radar observations of relative range and bearing. * could usefully be extended to include a discussion of the near-miss situation and the effect of observational errors.…”

“…* 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 5 and Morrell,6.7 and in graphical form by Wylie.8 The recent paper by Parker* 1 deals with the effects of both systematic and random errors of radar observations of relative range and bearing.…”

Dynamic Games Methods in Synthesis of Safe Ship Control Algorithms

“…Therefore, the management of such processes is accomplished by means of game control systems, whose synthesis is carried out with the methods of game theory [9,10]. Game theory is a branch of mathematics, covering the theory of conflict situations and building and analyzing their models [11,12]. Conflict can be as follows: military, political, social, and economic, in a social game, in the game with nature, and in the implementation of the control process during interferences of disturbances or other control objects [13].…”