14,183 research outputs found
Regular Moebius transformations of the space of quaternions
Let H be the real algebra of quaternions. The notion of regular function of a
quaternionic variable recently presented by G. Gentili and D. C. Struppa
developed into a quite rich theory. Several properties of regular quaternionic
functions are analogous to those of holomorphic functions of one complex
variable, although the diversity of the quaternionic setting introduces new
phenomena. This paper studies regular quaternionic transformations. We first
find a quaternionic analog to the Casorati-Weierstrass theorem and prove that
all regular injective functions from H to itself are affine. In particular, the
group Aut(H) of biregular functions on H coincides with the group of regular
affine transformations. Inspired by the classical quaternionic linear
fractional transformations, we define the regular fractional transformations.
We then show that each regular injective function from the Alexandroff
compactification of H to itself is a regular fractional transformation.
Finally, we study regular Moebius transformations, which map the unit ball B
onto itself. All regular bijections from B to itself prove to be regular
Moebius transformations.Comment: 12 page
Micro-to-macro: astrodynamics at extremes of lengths-scale
This paper investigates astrodynamics at extremes of length-scale, ranging from swarms of future `smart dust' devices to the capture and utilisation of small near Earth asteroids. At the smallest length-scales families of orbits are found which balance the energy gain from solar radiation pressure with energy dissipation due to air drag. This results in long orbit lifetimes for high area-to-mass ratio `smart dust' devices. High area-to-mass hybrid spacecraft, using both solar sail and electric propulsion, are then considered to enable `pole-sitter' orbits providing a polar-stationary vantage point for Earth observation. These spacecraft are also considered to enable displaced geostationary orbits. Finally, the potential material resource available from captured near Earth asteroids is considered which can underpin future large-scale space engineering ventures. The use of such material for geo-engineering is investigated using a cloud of unprocessed dust in the vicinity of the Earth-Sun point to fractionally reduce solar insolation
Density of high area-to-mass objects in Geostationary and Medium Earth orbits through semi-analytical equations and differential algebra
This paper introduces and combines two novel techniques. Firstly, we introduce an efficient numerical method for the propagation of entire sets of initial conditions in the phase space and their associated phase space densities based on Differential Algebra (DA) techniques. Secondly, this DA density propagator is applied to a DA-enabled implementation of Semi-Analytical (SA) averaged dynamics, combining for the first time the power of the SA and DA techniques. While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with SA equations yields a fast and accurate method to propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics. To demonstrate the effectiveness of the proposed method, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the effects of solar radiation pressure, the Earth’s oblateness and luni-solar perturbations. The computational efficiency is demonstrated by propagating 10; 000 random samples taking snapshots of their state and density at evenly spaced intervals throughout the integration. The total time required for a propagation for 16 years in the dynamics is on the order of tens of seconds on a common desktop PC
Electrochromic orbit control for smart-dust devices
Recent advances in MEMS (micro electromechanical systems) technology are leading to spacecraft which are the shape and size of computer chips, so-called SpaceChips, or ‘smart dust devices’. These devices can offer highly distributed sensing when used in future swarm applications. However, they currently lack a feasible strategy for active orbit control. This paper proposes an orbit control methodology for future SpaceChip devices which is based on exploiting the effects of solar radiation pressure using electrochromic coatings. The concept presented makes use of the high area-to-mass ratio of these devices, and consequently the large force exerted upon them by solar radiation pressure, to control their orbit evolution by altering their surface optical properties. The orbital evolution of Space Chips due to solar radiation pressure can be represented by a Hamiltonian system, allowing an analytic development of the control methodology. The motion in the orbital element phase space resembles that of a linear oscillator, which is used to formulate a switching control law. Additional perturbations and the effect of eclipses are accounted for by modifying the linearized equations of the secular change in orbital elements around an equilibrium point in the phase space of the problem. Finally, the effectiveness of the method is demonstrated in a test case scenario
Semi-autonomous Intersection Collision Avoidance through Job-shop Scheduling
In this paper, we design a supervisor to prevent vehicle collisions at
intersections. An intersection is modeled as an area containing multiple
conflict points where vehicle paths cross in the future. At every time step,
the supervisor determines whether there will be more than one vehicle in the
vicinity of a conflict point at the same time. If there is, then an impending
collision is detected, and the supervisor overrides the drivers to avoid
collision. A major challenge in the design of a supervisor as opposed to an
autonomous vehicle controller is to verify whether future collisions will occur
based on the current drivers choices. This verification problem is particularly
hard due to the large number of vehicles often involved in intersection
collision, to the multitude of conflict points, and to the vehicles dynamics.
In order to solve the verification problem, we translate the problem to a
job-shop scheduling problem that yields equivalent answers. The job-shop
scheduling problem can, in turn, be transformed into a mixed-integer linear
program when the vehicle dynamics are first-order dynamics, and can thus be
solved by using a commercial solver.Comment: Submitted to Hybrid Systems: Computation and Control (HSCC) 201
Quantum harmonic oscillator with superoscillating initial datum
In this paper we study the evolution of superoscillating initial data for the
quantum driven harmonic oscillator. Our main result shows that
superoscillations are amplified by the harmonic potential and that the analytic
solution develops a singularity in finite time. We also show that for a large
class of solutions of the Schr\"odinger equation, superoscillating behavior at
any given time implies superoscillating behavior at any other time.Comment: 12 page
Orbital dynamics of "smart dust" devices with solar radiation pressure and drag
This paper investigates how perturbations due to asymmetric solar radiation pressure, in the presence of Earth shadow, and atmospheric drag can be balanced to obtain long-lived Earth centred orbits for swarms of micro-scale 'smart dust' devices, without the use of active control. The secular variation of Keplerian elements is expressed analytically through an averaging technique. Families of solutions are then identified where Sun-synchronous apse-line precession is achieved passively to maintain asymmetric solar radiation pressure. The long-term orbit evolution is characterized by librational motion, progressively decaying due to the non-conservative effect of atmospheric drag. Long-lived orbits can then be designed through the interaction of energy gain from asymmetric solar radiation pressure and energy dissipation due to drag. In this way, the usual short drag lifetime of such high area-to-mass spacecraft can be greatly extended (and indeed selected). In addition, the effect of atmospheric drag can be exploited to ensure the rapid end-of-life decay of such devices, thus preventing long-lived orbit debris
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