On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.Comment: 20 page

Optimal randomized incremental construction for guaranteed logarithmic planar point location

Given a planar map of $n$ segments in which we wish to efficiently locate points, we present the first randomized incremental construction of the well-known trapezoidal-map search-structure that only requires expected $O(n \log n)$ preprocessing time while deterministically guaranteeing worst-case linear storage space and worst-case logarithmic query time. This settles a long standing open problem; the best previously known construction time of such a structure, which is based on a directed acyclic graph, so-called the history DAG, and with the above worst-case space and query-time guarantees, was expected $O(n \log^2 n)$. The result is based on a deeper understanding of the structure of the history DAG, its depth in relation to the length of its longest search path, as well as its correspondence to the trapezoidal search tree. Our results immediately extend to planar maps induced by finite collections of pairwise interior disjoint well-behaved curves.Comment: The article significantly extends the theoretical aspects of the work presented in http://arxiv.org/abs/1205.543

Complexity of the General Chromatic Art Gallery Problem

In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon $P$. We consider the Chromatic AGP (CAGP), where the guards are colored. As long as $P$ is completely covered, the number of guards does not matter, but guards with overlapping visibility regions must have different colors. This problem has applications in landmark-based mobile robot navigation: Guards are landmarks, which have to be distinguishable (hence the colors), and are used to encode motion primitives, \eg, "move towards the red landmark". Let $\chi_G(P)$, the chromatic number of $P$, denote the minimum number of colors required to color any guard cover of $P$. We show that determining, whether $\chi_G(P) \leq k$ is \NP-hard for all $k \geq 2$. Keeping the number of colors minimal is of great interest for robot navigation, because less types of landmarks lead to cheaper and more reliable recognition

Few-cycle Pulses Amplification For Attosecond Science Applications Modeling And Experiments

The emergence of mode-locked oscillators providing pulses with durations as short as a few electric-field cycles in the near infra-red has paved the way toward electric-field sensitive physics experiments. In addition, the control of the relative phase between the carrier and the pulse envelope, developed in the early 2000’s and rewarded by a Nobel price in 2005, now provides unprecedented control over the pulse behaviour. The amplification of such pulses to the millijoule level has been an on-going task in a few world-class laboratories and has triggered the dawn of attoscience, the science of events happening on an attosecond timescale. This work describes the theoretical aspects, modeling and experimental implementation of HERACLES, the Laser Plasma Laboratory optical parametric chirped pulse amplifier (OPCPA) designed to deliver amplified carrier-envelope phase stabilized 8-fs pulses with energy beyond 1 mJ at repetition rates up to 10 kHz at 800 nm central wavelength. The design of the hybrid fiber/solid-state amplifier line delivering 85-ps pulses with energy up to 10 mJ at repetition rates in the multi-kHz regime tailored for pumping the optical parametric amplifier stages is presented. The novel stretcher/compressor design of HERACLES, suitable for handling optical pulses with spectra exceeding 300 nm of bandwidth with unprecedented flexibility, is fully modeled and also presented in the frame of this thesis. Finally, a 3D model of the multistage non-collinear optical parametric amplifier is also reported. The current and foreseen overall performances of HERACLES are presented. This facility is designed to enable attosecond physics experiments, high-harmonic generation and physics of plasma studies

Exact computation of the adjacency graph of an arrangement of quadrics

Präsentiert wird ein vollständiger, exakter und effizienter Algorithmus zur Berechnung des Nachbarschaftsgraphen eines Arrangements von Quadriken (Algebraische Flächen vom Grad 2). Dies ist ein wichtiger Schritt auf dem Weg zur Berechnung des vollen 3D Arrangements. Dabei greifen wir auf eine bereits existierende Implementierung zur Berechnung der exakten Parametrisierung der Schnittkurve von zwei Quadriken zurück. Somit ist es möglich, die exakten Parameterwerte der Schnittpunkte zu bestimmen, diese entlang der Kurven zu sortieren und den Nachbarschaftsgraphen zu berechnen. Wir bezeichnen unsere Implementierung als vollständig, da sie auch die Behandlung aller Sonderfälle wie singulärer oder tangentialer Schnittpunkte einschließt. Sie ist exakt, da immer das mathematisch korrekte Ergebnis berechnet wird. Und schließlich bezeichnen wir unsere Implementierung als effizient, da sie im Vergleich mit dem einzigen bisher implementierten Ansatz gut abschneidet. Implementiert wurde unser Ansatz im Rahmen des Projektes EXACUS. Das zentrale Ziel von EXACUS ist es, einen Prototypen eines zuverlässigen und leistungsfähigen CAD Geometriekerns zu entwickeln. Obwohl wir das Design unserer Bibliothek als prototypisch bezeichnen, legen wir dennoch größten Wert auf Vollständigkeit, Exaktheit, Effizienz, Dokumentation und Wiederverwendbarkeit. Über den eigentlich Beitrag zu EXACUS hinaus, hatte der hier vorgestellte Ansatz durch seine besonderen Anforderungen auch wesentlichen Einfluss auf grundlegende Teile von EXACUS. Im Besonderen hat diese Arbeit zur generischen Unterstützung der Zahlentypen und der Verwendung modularer Methoden innerhalb von EXACUS beigetragen. Im Rahmen der derzeitigen Integration von EXACUS in CGAL wurden diese Teile bereits erfolgreich in ausgereifte CGAL Pakete weiterentwickelt.We present a complete, exact and efficient algorithm to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previously implemented approach. Our approach has been implemented within the EXACUS project. The central goal of Exacus is the development of a demonstrator of a reliable and efficient CAD geometry kernel. Although we call our library design prototypical, we spent nonetheless a great effort on completeness, exactness, efficiency, documentation and reusability. Beside its primary contribution, the algorithm presented in this work had an essential impact on fundamental parts of EXACUS due to its specific requirements. This work has in particular contributed to the generic number type support and the modular methods used within Exacus. In the context of our ongoing integration of EXACUS into CGAL these parts have been successfully advanced into mature CGAL packages

Geographieunterricht videobasiert analysieren – Konzeption und Evaluation eines videobasierten Lernsettings zur Förderung der professionellen Unterrichtswahrnehmung

Lehrpersonen werden im Unterrichtsalltag mit der Heterogenität von Lerngruppen konfrontiert. Um individuelle Lernangebote initiieren zu können, muss eine Lehrperson fähig sein, heterogenitäts- und lernrelevante Ereignisse im komplexen Unterrichtsgeschehen professionell wahrzunehmen. Zur Förderung dieser domänenspezifischen Fähigkeit bei angehenden Geographielehrkräften wurde eine videobasierte Lehrveranstaltung im Rahmen der vom BMBF geförderten Qualitätsoffensive Lehrerbildung an der Universität Münster konzipiert, durchgeführt und evaluiert. Der inhaltliche Analyseschwerpunkt liegt auf dem Einsatz von Lernunterstützungen als elementare Charakteristika eines heterogenitätssensiblen Unterrichts. Im Beitrag werden werden theorie- und empiriegestützt die Konzipierung, Durchführung und Evaluation der videobasierten Lehrveranstaltung vorgestellt

A Parallel Distributed Strategy for Arraying a Scattered Robot Swarm

We consider the problem of organizing a scattered group of $n$ robots in two-dimensional space, with geometric maximum distance $D$ between robots. The communication graph of the swarm is connected, but there is no central authority for organizing it. We want to arrange them into a sorted and equally-spaced array between the robots with lowest and highest label, while maintaining a connected communication network. In this paper, we describe a distributed method to accomplish these goals, without using central control, while also keeping time, travel distance and communication cost at a minimum. We proceed in a number of stages (leader election, initial path construction, subtree contraction, geometric straightening, and distributed sorting), none of which requires a central authority, but still accomplishes best possible parallelization. The overall arraying is performed in $O(n)$ time, $O(n^2)$ individual messages, and $O(nD)$ travel distance. Implementation of the sorting and navigation use communication messages of fixed size, and are a practical solution for large populations of low-cost robots