The knowledge of knots: an interdisciplinary literature review

ABSTRACT

Knots can be found and used in a variety of situations in the 3D world, such as in vines, in the DNA, polymer chains, electrical wires, in mountaineering, seamanship and when ropes or other flexible objects are involved for exerting forces and holding objects in place. Research on knots as topological entities has contributed with a number of findings, not only of interest to pure mathematics, but also to statistical mechanics, quantum physics, genetics, and chemistry. Yet, the cognitive (or algorithmic) aspects involved in the act of tying a knot are a largely uncharted territory. This paper presents a review of the literature related to the investigation of knots from the topological, physical, cognitive and computational (including robotics) standpoints, aiming at bridging the gap between pure mathematical work on knot theory and macroscopic physical knots, with an eye to applications and modeling.     

The GANDER constellation for maritime dissaster mitigation

The development of sea state monitoring from polar-orbiting satellites has recently moved away from the concept of single, multi-sensor platform such as ERS-2, Topex/Poseidon or ENVISAT towards the design of a system that would allow frequent updates from a constellation of small satellites equipped with special-purpose radar altimeters. This new system, called GANDER for Global Altimeter Network Designed to Evaluate Risk, has attracted significant support from a number of important customer segments including the military.

This paper details the design of an altimeter for a Surrey small satellite, and illustrates the major system trade-offs that need to be made. Critical to the viability of the mission will be the development of a radar altimeter capable of operating successfully on a small satellite bus, within a limited volume and power budget. The mission design presents a number of key technological challenges, in order to permit a physically small antenna to be employed, and to minimise the pulse power. This can be achieved by advanced techniques, such as the delay Doppler altimeter concept, which emphasises the needs for high-speed on-board signal processing, phase linearity and pulse-to-pulse phase coherency.

The system design for the GANDER constellation is also described, illustrating how it not only offers a means for maritime disaster mitigation, but also can reduce shipping cost and time.     

In-flight testing of the injection of the LISA Pathfinder test mass into a geodesic

LISA Pathfinder is a technology demonstrator space mission, aimed at testing key technologies for detecting gravitational waves in space. The mission is the precursor of LISA, the first space gravitational waves observatory, whose launch is scheduled for 2034. The LISA Pathfinder scientific payload includes two gravitational reference sensors (GRSs), each one containing a test mass (TM), which is the sensing body of the experiment. A mission critical task is to set each TM into a pure geodesic motion, i.e. guaranteeing an extremely low acceleration noise in the sub-Hertz frequency bandwidth. The grabbing positioning and release mechanism (GPRM), responsible for the injection of the TM into a geodesic trajectory, was widely tested on ground, with the limitations imposed by the 1-g environment. The experiments showed that the mechanism, working in its nominal conditions, is capable of releasing the TM into free-fall fulfilling the very strict constraint imposed on the TM residual velocity, in order to allow its capture on behalf of the electrostatic actuation.However, the first in-flight releases produced unexpected residual velocity components, for both the TMs. Moreover, all the residual velocity components were greater than maximum value set by the requirements. The main suspect is that unexpected contacts took place between the TM and the surroundings bodies. As a consequence, ad hoc manual release procedures had to be adopted for the few following injections performed during the nominal mission. These procedures still resulted in non compliant TM states which were captured only after impacts. However, such procedures seem not practicable for LISA, both for the limited repeatability of the system and for the unmanageable time lag of the telemetry/telecommand signals (about 4400 s). For this reason, at the end of the mission, the GPRM was deeply tested in-flight, performing a large number of releases, according to different strategies. The tests were carried out in order to understand the unexpected dynamics and limit its effects on the final injection. Some risk mitigation maneuvers have been tested aimed at minimizing the vibration of the system at the release and improving the alignment between the mechanism and the TM. However, no overall optimal release strategy to be implemented in LISA could be found, because the two GPRMs behaved differently.     

Transfer orbits to/from the Lagrangian points in the restricted four-body problem

The well-known Lagrangian points that appear in the planar restricted three-body problem are very important for astronautical applications. They are five points of equilibrium in the equations of motion, what means that a particle located at one of those points with zero velocity will remain there indefinitely. The collinear points (L₁, L₂ and L₃) are always unstable and the triangular points (L₄ and L₅) are stable in the present case studied (Earth–Sun system). They are all very good points to locate a space-station, since they require a small amount of ΔV (and fuel), the control to be used, for station-keeping. The triangular points are especially good for this purpose, since they are stable equilibrium points.In this paper, the planar restricted four-body problem applied to the Sun–Earth–Moon–Spacecraft is combined with numerical integration and gradient methods to solve the two-point boundary value problem. This combination is applied to the search of families of transfer orbits between the Lagrangian points and the Earth, in the Earth–Sun system, with the minimum possible cost of the control used. So, the final goal of this paper is to find the magnitude of the two impulses to be applied in the spacecraft to complete the transfer: the first one when leaving/arriving at the Lagrangian point and the second one when arriving/living at the Earth.The dynamics given by the restricted four-body problem is used to obtain the trajectory of the spacecraft, but not the position of the equilibrium points. Their position is taken from the restricted three-body model. The goal to use this model is to evaluate the perturbation of the Sun in those important trajectories, in terms of fuel consumption and time of flight. The solutions will also show how to apply the impulses to accomplish the transfers under this force model.The results showed a large collection of transfers, and that there are initial conditions (position of the Sun with respect to the other bodies) where the force of the Sun can be used to reduce the cost of the transfers.