Application of Difference Schemes to Decision the Pursuit Problem

The problem of the pursuit curve construction in the case when the tangent to pursuer’s motion trajectory passes at any time through the point representing the pursued is considered. A new approach to construct the pursuit curves using difference schemes is proposed. The proposed technique eliminates the need to derive the differential equations for the description of the pursuit curves, which is quite difficult task in the general case. In addition, the application of difference methods is justified in a situation where it is complicated to find the analytical solution of an existing differential equation and it is possible to obtain the pursuit curve only numerically. Various modifications of difference schemes respectively equivalent to the Euler, to the Adams – Bashforth and to the Milne methods are constructed. Their software implementation is realized by using the mathematical package Mathcad. We consider the case of a uniform rectilinear motion of the pursued whose differential equation describing the path of the pursuer and its analytical solution are known. We compare the numerical solutions obtained by the different methods with the well-known analytical solution. The error of the obtained numerical solutions is examined. Moreover, an application is considered illustrating the construction of the difference schemes for the case of an arbitrary trajectory of the pursued. Also, we extend the proposed method to the case of cyclic pursuit with several participants in the three-dimensional space. In particular, we construct a difference scheme equivalent to the Euler method for a three-dimensional analogue of the "bugs problem". The results obtained are demonstrated by means of animated examples for either two-dimensional or three-dimensional cases.

Tractrix Trajectory with Slip Steering

The tractrix curve, sometimes called the pursuit curve has long been the standard used to describe the path of a pig trailer behind a prime mover. This ideal path still has validity today provided the speed is very low and the trailer is unloaded. During a common phenomenon of snaking or fishtailing, the trailer sways back and forth in relation to the prime mover centreline axis. Often regarded as the nightmare of caravanning, the action does not follow the tractrix curve but follows a shorter path to the common centreline of prime mover and trailer. This paper explores the shorter path in response to a tyre reaction to centripetal force causing slip steer. An example derived by drafting progression steps to how quantitatively that speed causes early cross-over carrying more energy into the next fishtailing phase is presented. It is believed the inclusion of slip steering to modify a tractrix curve is a novel development.

GEOMETRY OF VIBRATIONAL STABILIZATION AND SOME APPLICATIONS

This paper gives a short overview of various applications of stabilization by vibration, along with the exposition of the geometrical mechanism of this phenomenon. More specifically, the following observation is described: a rapidly vibrated holonomic system can be approximated by a certain associated nonholonomic system. It turns out that effective forces in some rapidly vibrated (holonomic) systems are the constraint forces of an associated auxiliary nonholonomic constraint. In particular, we review a simple but remarkable connection between the curvature of the pursuit curve (the tractrix) on the one hand and the effective force on the pendulum with vibrating support. The latter observation is a part of a recently discovered close relationship between two standard classical problems in mechanics: (1) the pendulum whose suspension point executes fast periodic motion along a given curve, and (2) the Chaplygin skate (known also as the Prytz planimeter, or the "bicycle"). The former is holonomic, the latter is nonholonomic. The holonomy of the skate shows up in the effective motion of the pendulum. This relationship between the pendulum with a twirled pivot and the Chaplygin skate has somewhat unexpected physical manifestations, such as the drift of suspended particles in acoustic waves. Finally, a higher-dimensional example of "geodesic motion" on a vibrating surface is described.

The Air-to-air Missile Matra 530

The Matra 530 is currently called a second generation air-to-air missile. Indeed the first generation of air-to-air missiles was represented in France by the Nord 5103 which was command-guided, or the Matra 511 which was a semi-active, pure pursuit curve missile. In the second part of the 1950's the requirement was obvious for a sophisticated interception system capable of coping with the high and medium altitude threat represented by heavy and medium bombers. The destruction of this threat required a close link with an interception system composed of ground radars, ground control equipment, sophisticated high performance interceptor aircraft and, as a last link, a very high performance missile. From the French Air Force point of view, the two main characteristics of this missile as the last link of an interceptor system, were first its flexibility in different weather environment and counter measure environment; second, its capability to decrease to the maximum the penetration of the enemy bomber, which meant that the missile should be capable of attacking not only in the rear sector but also in frontal attack and in fact in all sectors around the attacking bomber.