Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

Pseudo-Triangulations, Rigidity and Motion Planning

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

Reconfiguring Colloidal Solids with Defects Using Active Matter

Engineering defect configurations within atomic crystalline materials, particularly metals, is a cornerstone of material science. Crystalline defects affect every facet of a material's properties, and in this regard, crystals composed of colloidal particles are no different from their atomic analogues. What is different, at the colloidal scale, is that new techniques have been developed to allow for the application of local forces by independently operating um-scale particles. The capacity of such active matter to manipulate, produce, and remove colloidal defects is only just starting to be explored. This dissertation seeks to establish the feasibility of directly controlling crystalline defects through the action of a particle species that exerts work locally, via computer simulation. The goals of such microstructure manipulation are to create materials with dynamic properties that change in response to external stimuli. Dynamic modification of crystallite shape, optical properties, or mechanical properties such as resistance to deformation are examples of what can be achieved through the action of active particles strongly coupled to colloidal defects. This dissertation is built around four studies of the behavior of defects in colloidal materials. First, I examine the nature of dislocations in crystals composed of particles interacting through repulsive pair potentials. By comparing attractive potentials to a family of repulsive ones with differing slopes, I explore the changes to mechanical properties and dislocation structure that occur as entropy comes to dominate the deformation free energy of a material. By varying the confining pressure, I find that attractive and repulsive systems can be matched in material properties and defect strain fields. Second, I study the interactions between colloidal dislocations and anisotropic interstitial particles that are capable of exerting local forces. By representing this interstitial by the strain field it produces when embedded in the crystal, I formulate a method of optimizing the interaction of the dislocation and interstitial by allowing the strain field to fluctuate. The optimization can be carried out very quickly compared to schemes requiring molecular dynamics simulation to assess a trial geometry's fitness. By molecular dynamics simulation of optimized particles with dislocations I show that such defects can be induced to glide by the action of bound active interstitials. Third, I explore the interaction of active rod-like interstitial particles with stacking faults in face-centered cubic colloidal crystals of repulsive spheres. I find that certain geometries of active interstitials are capable of efficiently searching through a crystal and binding strongly to a stacking fault. They rapidly encounter the stacking faults that link partial dislocations in the FCC crystal, and when absorbed provide an additional barrier to dislocation glide. The presence of such optimized active, stacking-fault seeking interstitials can be detected in the shear deformation properties of dislocated crystals even at concentrations as small as 64 per million host particles. Fourth, I explore how a crystalline colloidal robot could be reconfigured using shear displacements resulting from the biased migration of dislocations. I propose a means of creating and controlling the migration of dislocations in 2D colloidal crystals based on embedded clusters of particles capable of changing size. I show that for clusters of particular geometries, cyclic expansion and contraction of their constituent particles produce dislocations that accumulate slip. Single or multiple slip planes can be used to reshape the boundaries of a 2D colloidal crystallite.PHDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/153451/1/bvansade_1.pd

Straight Line Movement in Morphing and Pursuit Evasion

Piece-wise linear structures are widely used to define problems and to represent simplified solutions in computational geometry. A piece-wise linear structure consists of straight-line or linear pieces connected together in a continuous geometric environment like 2D or 3D Euclidean spaces. In this thesis two different problems both with the approach of finding piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion and straight-line morphing. Straight-line pursuit evasion is a geometric version of the famous cops and robbers game that is defined in this thesis for the first time. The game is played in a simply connected region in 2D. It is a full information game where the players take turns. The cop’s goal is to catch the robber. In a turn, each player may move any distance along a straight line as long as the line segment connecting their current location to the new location is not blocked by the region’s boundary. We first prove that the cop can always win the game when the players move on the visibility graph of a simple polygon. We prove this by showing that the visibility graph of a simple polygon is “dismantlable” (the known class of cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other settings of the game are also studied in this thesis: when the players are free to move on the infinitely many points inside a simple polygon, and inside a splinegon. In both cases we show that the cop can always win the game. For the case of polygons, the proposed cop strategy gives an asymptotically tight linear bound on the number of steps the cop needs to catch the robber. For the case of splinegons, the cop may need a quadratic number of steps with the proposed strategy, while our best lower bound is linear. Straight-line morphing is a type of morphing first defined in this thesis that provides a nice and smooth transformation between straight-line graph drawings in 2D. In straight- line morphing, each vertex of the graph moves forward along the line segment connecting its initial position to its final position. The vertex trajectories in straight-line morphing are very simple, but because the speed of each vertex may vary, straight-line morphs are more general than the commonly used “linear morphs” where each vertex moves at uniform speed. We explore the problem of whether an initial planar straight-line drawing of a graph can be morphed to a final straight-line drawing of the graph using a straight-line morph that preserves planarity at all times. We prove that this problem is NP-hard even for the special case where the graph drawing consists of disjoint segments. We then look at some restricted versions of the straight-line morphing: when only one vertex moves at a time, when the vertices move one by one to their final positions uninterruptedly, and when the edges morph one by one to their final configurations in the case of disjoint segments. Some of the variations are shown to be still NP-complete while some others are solvable in polynomial time. We conjecture that the class of planar straight-line morphs is as powerful as the class of planar piece-wise linear straight-line morphs. We also explore a simpler problem where for each edge the quadrilateral formed by its initial and final positions together with the trajectories of its two vertices is convex. There is a necessary condition for this case that we conjecture is also sufficient for paths and cycles

Proceedings of the NASA Conference on Space Telerobotics, volume 4

Papers presented at the NASA Conference on Space Telerobotics are compiled. The theme of the conference was man-machine collaboration in space. The conference provided a forum for researchers and engineers to exchange ideas on the research and development required for the application of telerobotic technology to the space systems planned for the 1990's and beyond. Volume 4 contains papers related to the following subject areas: manipulator control; telemanipulation; flight experiments (systems and simulators); sensor-based planning; robot kinematics, dynamics, and control; robot task planning and assembly; and research activities at the NASA Langley Research Center

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Mid-Century Molecular: The Material Culture of X-ray Crystallographic Visualisation across Postwar British Science and Industrial Design

This thesis investigates the use and significance of X-ray crystallographic visualisations of molecular structures in postwar British material culture across scientific practice and industrial design. It is based on research into artefacts from three areas: X-ray crystallographers’ postwar practices of visualising molecular structures using models and diagrams; the Festival Pattern Group scheme for the 1951 Festival of Britain, in which crystallographic visualisations formed the aesthetic basis of patterns for domestic objects; and postwar furnishings with a ‘ball-and-rod’ form and construction reminiscent of those of molecular models. A key component of the project is methodological. The research brings together subjects, themes and questions traditionally covered separately by two disciplines, the history of design and history of science. This focus necessitated developing an interdisciplinary set of methods, which results in the reassessment of disciplinary borders and productive cross-disciplinary methodological applications. This thesis also identifies new territory for shared methods: it employs network models to examine cross-disciplinary interaction between practitioners in crystallography and design, and a biographical approach to designed objects that over time became mediators of historical narratives about science. Artefact-based, archival and oral interviewing methods illuminate the production, use and circulation of the objects examined in this research. This interdisciplinary approach underpins the generation of new historical narratives in this thesis. It revises existing histories of the cultural transmissions between X-ray crystallography and the production and reception of designed objects in postwar Britain. I argue that these transmissions were more complex than has been acknowledged by historians: they were contingent upon postwar scientific and design practices, material conditions in postwar Britain and the dynamics of historical memory, both scholarly and popular. This thesis comprises four chapters. Chapter one explores X-ray crystallographers’ visualisation practices, conceived here as a form of craft. Chapter two builds on this, demonstrating that the Festival Pattern Group witnesses the encounter between crystallographic practice, design practice and aesthetic ideologies operating within social networks associated with postwar modernisms. Chapters three and four focus on ball-and-rod furnishings in postwar and present-day Britain, respectively. I contend that strong relationships between these designed objects and crystallographic visualisations, for example the appellation ‘atomic design’, have been largely realised through historical narratives active today in the consumption of ‘retro’ and ‘mid-century modern’ artefacts. The attention to contemporary historical narratives necessitates this dual historical focus: the research is rooted in the period from the end of the Second World War until the early 1960s, but extends to the history of now. This thesis responds to the need for practical research on methods for studying cross-disciplinary interactions and their histories. It reveals the effects of submitting historical subjects that are situated on disciplinary boundaries to interdisciplinary interpretation. Old models, such as that of unidirectional ‘influence’, subside and the resulting picture is a refracted one: this study demonstrates that the material form and meaning of crystallographic visualisations, within scientific practice and across their use and echoes in designed objects, are multiple and contingent

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum