Optimal circular flight of multiple UAVs for target tracking in urban areas

This work is an extension of our previous result in which a novel single-target tracking algorithm for fixed-wing UAVs (Unmanned Air Vehicles) was proposed. Our previous algorithm firstly finds the centre of a circular flight path, rc, over the interested ground target which maximises the total chance of keeping the target inside the camera field of view of UAVs, , while the UAVs fly along the circular path. All the UAVs keep their maximum allowed altitude and fly along the same circle centred at rc with the possible minimum turn radius of UAVs. As discussed in [1,4], these circular flights are highly recommended for various target tracking applications especially in urban areas, as for each UAV the maximum altitude flight ensures the maximum visibility and the minimum radius turn keeps the minimum distance to the target at the maximum altitude. Assuming a known probability distribution for the target location, one can quantify , which is incurred by the travel of a single UAV along an arbitrary circle, using line-of-sight vectors. From this observation, (the centre of) an optimal circle among numerous feasible ones can be obtained by a gradient-based search combined with random sampling, as suggested in [1]. This optimal circle is then used by the other UAVs jointly tracking the same target. As the introduction of multiple UAVs may minimise further, the optimal spacing between the UAVs can be naturally considered. In [1], a typical line search method is suggested for this optimal spacing problem. However, as one can easily expect, the computational complexity of this search method may undesirably increase as the number of UAVs increases. The present work suggests a remedy for this seemingly complex optimal spacing problem. Instead of depending on time-consuming search techniques, we develop the following algorithm, which is computationally much more efficient. Firstly, We calculate the distribution (x), where x is an element of , which is the chance of capturing the target by one camera along . Secondly, based on the distribution function, (x), find separation angles between UAVs such that the target can be always tracked by at least one UAV with a guaranteed probabilistic measure. Here, the guaranteed probabilistic measure is chosen by taking into account practical constraints, e.g. required tracking accuracy and UAVs' minimum and maximum speeds. Our proposed spacing scheme and its guaranteed performance are demonstrated via numerical simulations

Coupled oscillators and Feynman's three papers

According to Richard Feynman, the adventure of our science of physics is a perpetual attempt to recognize that the different aspects of nature are really different aspects of the same thing. It is therefore interesting to combine some, if not all, of Feynman's papers into one. The first of his three papers is on the ``rest of the universe'' contained in his 1972 book on statistical mechanics. The second idea is Feynman's parton picture which he presented in 1969 at the Stony Brook conference on high-energy physics. The third idea is contained in the 1971 paper he published with his students, where they show that the hadronic spectra on Regge trajectories are manifestations of harmonic-oscillator degeneracies. In this report, we formulate these three ideas using the mathematics of two coupled oscillators. It is shown that the idea of entanglement is contained in his rest of the universe, and can be extended to a space-time entanglement. It is shown also that his parton model and the static quark model can be combined into one Lorentz-covariant entity. Furthermore, Einstein's special relativity, based on the Lorentz group, can also be formulated within the mathematical framework of two coupled oscillators.Comment: 31 pages, 6 figures, based on the concluding talk at the 3rd Feynman Festival (Collage Park, Maryland, U.S.A., August 2006), minor correction

Einstein's Hydrogen Atom

In 1905, Einstein formulated his special relativity for point particles. For those particles, his Lorentz covariance and energy-momentum relation are by now firmly established. How about the hydrogen atom? It is possible to perform Lorentz boosts on the proton assuming that it is a point particle. Then what happens to the electron orbit? The orbit could go through an elliptic deformation, but it is not possible to understand this problem without quantum mechanics, where the orbit is a standing wave leading to a localized probability distribution. Is this concept consistent with Einstein's Lorentz covariance? Dirac, Wigner, and Feynman contributed important building blocks for understanding this problem. The remaining problem is to assemble those blocks to construct a Lorentz-covariant picture of quantum bound states based on standing waves. It is shown possible to assemble those building blocks using harmonic oscillators.Comment: LaTex 15 pages, 5 figures, presented at the International Workshop on Physics and Mathematics (Hangzhou, China, July 2011), to be published in the procedding

Group Contractions: Inonu, Wigner, and Einstein

Einstein's $E = mc^{2}$ unifies the momentum-energy relations for massive and massless particles. According to Wigner, the internal space-time symmetries of massive and massless particles are isomorphic to $O(3)$ and $E(2)$ respectively. According to Inonu and Wigner, $O(3)$ can be contracted to $E(2)$ in the large-radius limit. It is noted that the $O(3)$-like little group for massive particles can be contracted to the $E(2)$-like little group for massless particles in the limit of large momentum and/or small mass. It is thus shown that transverse rotational degrees of freedom for massive particles become contracted to gauge degrees of freedom for massless particles.Comment: 9 pages, LaTeX file, no figures; presented at the 2nd International Workshop on Classical and Quantum Integrable Systems: Algebraic Methods and Lie Algebra Contractions (Dubna, July 8-12, 1996), to be published in the Proceeding

Quantum-disordered slave-boson theory of underdoped cuprates

We study the stability of the spin gap phase in the U(1) slave-boson theory of the t-J model in connection to the underdoped cuprates. We approach the spin gap phase from the superconducting state and consider the quantum phase transition of the slave-bosons at zero temperature by introducing vortices in the boson superfluid. At finite temperatures, the properties of the bosons are different from those of the strange metal phase and lead to modified gauge field fluctuations. As a result, the spin gap phase can be stabilized in the quantum critical and quantum disordered regime of the boson system. We also show that the regime of quantum disordered bosons with the paired fermions can be regarded as the strong coupling version of the recently proposed nodal liquid theory.Comment: 5 pages, Replaced by the published versio