Optimal rendezvous on a line by location-aware robots in the presence of spies

A set of mobile robots is placed at arbitrary points of an infinite line. The robots are equipped with GPS devices and they may communicate their positions on the line to a central authority. The collection contains an unknown subset of “spies”, i.e., byzantine robots, which are indistinguishable from the non-faulty ones. The set of the non-faulty robots needs to rendezvous in the shortest possible time in order to perform some task, while the byzantine robots may try to delay their rendezvous for as long as possible. The problem facing a central authority is to determine trajectories for all robots so as to minimize the time until all the non-faulty robots have met. The trajectories must be determined without knowledge of which robots are faulty. Our goal is to minimize the competitive ratio between the time required to achieve the first rendezvous of the non-faulty robots and the time required for such a rendezvous to occur under the assumption that the faulty robots are known at the start. In this paper, we give rendezvous algorithms with bounded competitive ratio, where the central authority is informed only of the set of initial robot positions, without knowing which ones or how many of them are faulty. In general, regardless of the number of faults [Formula: see text] it can be shown that there is an algorithm with bounded competitive ratio. Further, we are able to give a rendezvous algorithm with optimal competitive ratio provided that the number [Formula: see text] of faults is strictly less than [Formula: see text]. Note, however, that in general this algorithm does not give an estimate on the actual value of the competitive ratio. However, when an upper bound on the number of byzantine robots is known to the central authority, we can provide algorithms with constant competitive ratios and in some instances we are able to show that these algorithms are optimal. Moreover, in the cases where the number of faults is either [Formula: see text] or [Formula: see text] we are able to compute the competitive ratio of an optimal rendezvous algorithm, for a small number of robots.

On a nonlinear problem of optimal rendezvous

The paper analyzes a nonlinear problem of optimal rendezvous of two material points in the horizontal plane. The velocity of both participants is constant modulo. The aim of control is to minimize the final distance between participants under given initial conditions. The approach time is fixed. The angle between the line of sight and the velocity vector of the Participant 1 (P1) is used as a control variable. The Participant (P2) uses the proportional-navigation law. This task may be relevant when planning the approach paths of a tanker aircraft to an unmanned aerial vehicle, or in the case of intercepting an attacking unmanned aerial vehicle by a target simulator missile launched from a real target. The principle of maximum procedure allows reducing optimal control problem to the problem of analyzing the phase portrait of a system of two nonlinear differential equations. A qualitative analysis of the system is performed, the characteristic properties of the trajectories of the participants in the horizontal plane are investigated and the results of numerical solution of the boundary value problem are presented.

Transfers with a rendezvous lasting one to two orbits between near-circular coplanar orbits

The paper discusses orbit transfers involving spacecraft rendezvous which belong to the class of coplanar non-intersecting orbits of a spacecraft and a space station. The duration of the rendezvous is assumed to be limited to two orbits, because for longer durations there is a known optimal solution algorithm, where phasing is achieved through the optimal orbit-to-orbit transfer between coplanar orbits. The proposed programs include in the final leg of the transfer a three-impulse rendezvous program lasting one orbit, which was determined using the method of splitting the impulse burns for the optimal orbit-to-orbit transfer. The phasing needed to achieve the phase difference required at the start of the three-impulse rendezvous program is attained through maneuvering during the previous leg of the transfer and the splitting of the orbit transfer pulses, not resulting in an increased propellant consumption. The structure of the optimal rendezvous program as function of its duration was determined and computing formulas were obtained. The range of phase differences at the start of maneuvering was determined, within which the characteristic velocity of the rendezvous is equal to the characteristic velocity of the orbit-to-orbit transfer. The paper presents simulation results for “quick" rendezvous profiles that use the proposed programs. Key words: spacecraft, orbital station, «quick» rendezvous, orbit transfer, rendezvous program.