A 2-chain can interlock with a k-chain

One of the open problems posed in [3] is: what is the minimal number k such that an open, flexible k-chain can interlock with a flexible 2-chain? In this paper, we establish the assumption behind this problem, that there is indeed some k that achieves interlocking. We prove that a flexible 2-chain can interlock with a flexible, open 16-chain.Comment: 10 pages, 6 figure

A 2-chain can interlock with an open 10-chain

It is an open problem, posed in \cite{SoCG}, to determine the minimal $k$ such that an open flexible $k$-chain can interlock with a flexible 2-chain. It was first established in \cite{GLOSZ} that there is an open 16-chain in a trapezoid frame that achieves interlocking. This was subsequently improved in \cite{GLOZ} to establish interlocking between a 2-chain and an open 11-chain. Here we improve that result once more, establishing interlocking between a 2-chain and a 10-chain. We present arguments that indicate that 10 is likely the minimum.Comment: 9 pages, 6 figure

Any Monotone Function Is Realized by Interlocked Polygons

Suppose there is a collection of n simple polygons in the plane, none of which overlap each other. The polygons are interlocked if no subset can be separated arbitrarily far from the rest. It is natural to ask the characterization of the subsets that makes the set of interlocked polygons free (not interlocked). This abstracts the essence of a kind of sliding block puzzle. We show that any monotone Boolean function ƒ on n variables can be described by m = O(n) interlocked polygons. We also show that the decision problem that asks if given polygons are interlocked is PSPACE-complete

Locked and unlocked smooth embeddings of surfaces

We study the continuous motion of smooth isometric embeddings of a planar surface in three-dimensional Euclidean space, and two related discrete analogues of these embeddings, polygonal embeddings and flat foldings without interior vertices, under continuous changes of the embedding or folding. We show that every star-shaped or spiral-shaped domain is unlocked: a continuous motion unfolds it to a flat embedding. However, disks with two holes can have locked embeddings that are topologically equivalent to a flat embedding but cannot reach a flat embedding by continuous motion.Comment: 8 pages, 8 figures. To appear in 34th Canadian Conference on Computational Geometr

THE ANTHROPOMETRY OF BODY ACTION

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73105/1/j.1749-6632.1955.tb32112.x.pd

Sliceforms: Deployable structures from interlocking slices

A sliceform is a volumetric, honeycomb-like structure assembled from an array of cross-sectional planar slices that are interlocked via pairs of complementary slots placed along each intersection. If the slices are thin, these slotted intersections function as revolute joints, and the sliceform is foldable if the geometry of the embedded spatial linkage permits it, for example a lattice sliceform (LS) is bi-directionally flat-foldable. This thesis concerns a study of such sliceforms toward the design of novel deployable structures. A sliceform torus, composed of two sets of inclined slices arranged at regular intervals about a central axis of symmetry, has been discovered to exhibit a surprising and intriguing folding action whereby its incomplete form can be collapsed to a flat-folded stack of coplanar slices. On deployment, the assembly expands smoothly about an arc until the slices have rotated to their design inclination, then, without reaching any apparent physical limit, abruptly ‘locks out’. With a full complement of slices, the outermost intersections can be interlocked to complete and rigidify the ring. The torus is an example of a rotational sliceform (RS), and analysis of these structures proceeds by noting that their structural geometry comprises an array of pyramidal cells that is commensurate to a spherical scissor grid. The conditions for flat-foldability are determined by examination of the intrinsic geometry of each cell; the incompatibility of the slices with apparent rigid-folding revealed by assessment of the extrinsic motion of the slices. Investigation of their compliant kinematics reveals the articulation to be a bistable transition admitted by small transverse deflections of the slices. This structural form is generalised by development of a technique for generating sliceforms along a smooth spatial curve – curve sliceforms (CS). Their synthesis is more involved than for an RS, but a range of sliceform ‘tubes’ are generated and manufactured. Each example retains the flat-foldable, deployable characteristic of an RS, despite the apparent intrinsic rigidity of each constituent skew cell. Examination of the small-scale models indicates that deployable motion is achieved via imperfect action of the slots, and a simple model of the articulation of a single cell is constructed to investigate how this proceeds, verifying that motion is kinematically admissible via local deformations

Real-time Control of Robot Arm Based on Hand Tracking Using Leap Motion Sensor Technology

Leap Motion is an example of ground-breaking technology that has the potential to change the way we control machines and therefore, how we control our world! Leap Motion is a sensor that is currently used to navigate through a personal computer with just hand gestures. For our MQP, the team has used this technology to control a physical robot arm. For the second part of this project, the team used SolidWorks to design a six-degree-of–freedom robot arm with five human-like fingers. The robot was designed to be controlled by Leap Motion, with human hand gestures as the input. It utilizes all of the sensor’s features, including the simultaneous control of all five fingers. The robot could be used for virtually any application, including service in the medical or military fields

Digital Material Assembly by Passive Means and Modular Isotropic Lattice Extruder System

A set of machines and related systems build structures by the additive assembly of discrete parts. These digital material assemblies constrain the constituent parts to a discrete set of possible positions and orientations. In doing so, the structures exhibit many of the properties inherent in digital communication such as error correction, fault tolerance and allow the assembly of precise structures with comparatively imprecise tools. Assembly of discrete cellular lattices by a Modular Isotropic Lattice Extruder System (MILES) is implemented by pulling strings of lattice elements through a forming die that enforces geometry constraints that lock the elements into a rigid structure that can then be pushed against and extruded out of the die as an assembled, loadbearing structure

A Foundation for Analysis of Spherical System Linkages Inspired by Origami and Kinematic Paper Art

Origami and its related fields of paper art are known to map to mechanisms, permitting kinematic analysis. Many origami folds have been studied in the context of engineering applications, but a sufficient foundation of principles of the underlying class of mechanism has not been developed. In this work, the mechanisms underlying paper art are identified as “spherical system linkages” and are studied in the context of generic mobility analysis with the goal of establishing a foundation upon which future work can develop.Spherical systems consist of coupled spherical and planar loops, and they motivate a reclassification of mechanisms based on the Chebyshev-Grübler-Kutzbach framework. Spherical systems are capable of complex, closed-loop motion in 3D space despite the mobility calculation treating the links as constrained to a single 2D surface. This property permits generalization of some multi-loop planar mechanisms, such as the Watt mechanism, to a generalized 3D form with equal mobility. A minimal connectivity graph representation of spherical systems is developed, and generic mobility equations are identified. Spherical system linkages are generalized further into spherical/spatial hybrid mechanisms which may have any combination of spherical, planar, and spatial loops. These are represented and analyzed with a polyhedron model. The connectivity graph is modified for this case and appropriate generic mobility equations are identified and adapted.The generic analyses developed for spherical system linkages are sufficient to inform an exhaustive type synthesis process. Generation of all configurations of a paper art inspired mechanism subject to constraints is discussed, and a case study generates all configurations of a spatial chain using specified link types. This design process is enabled by the developed notation and analyses, which are used to identify, depict, and classify kinematic paper art inspired mechanisms