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Register a new userFolding a Paper Strip to Minimize Thickness
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Added by
Yushi Uno
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a flat folding is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where thicker creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers.
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Given a set of objects with durations (jobs) that cover a base region, can we schedule the jobs to maximize the duration the original region remains covered? We call this problem the sensor cover problem. This problem arises in the context of covering a region with sensors. For example, suppose you wish to monitor activity along a fence by sensors placed at various fixed locations. Each sensor has a range and limited battery life. The problem is to schedule when to turn on the sensors so that the fence is fully monitored for as long as possible. This one dimensional problem involves intervals on the real line. Associating a duration to each yields a set of rectangles in space and time, each specified by a pair of fixed horizontal endpoints and a height. The objective is to assign a position to each rectangle to maximize the height at which the spanning interval is fully covered. We call this one dimensional problem restricted strip covering. If we replace the covering constraint by a packing constraint, the problem is identical to dynamic storage allocation, a scheduling problem that is a restricted case of the strip packing problem. We show that the restricted strip covering problem is NP-hard and present an O(log log n)-approximation algorithm. We present better approximations or exact algorithms for some special cases. For the uniform-duration case of restricted strip covering we give a polynomial-time, exact algorithm but prove that the uniform-duration case for higher-dimensional regions is NP-hard. Finally, we consider regions that are arbitrary sets, and we present an O(log n)-approximation algorithm.
Given n jobs with release dates, deadlines and processing times we consider the problem of scheduling them on m parallel machines so as to minimize the total energy consumed. Machines can enter a sleep state and they consume no energy in this state. Each machine requires Q units of energy to awaken from the sleep state and in its active state the machine can process jobs and consumes a unit of energy per unit time. We allow for preemption and migration of jobs and provide the first constant approximation algorithm for this problem.
Accurate manipulation of a deformable body such as a piece of fabric is difficult because of its many degrees of freedom and unobservable properties affecting its dynamics. To alleviate these challenges, we propose the application of feedback-based control to robotic fabric strip folding. The feedback is computed from the low dimensional state extracted from a camera image. We trained the controller using reinforcement learning in simulation which was calibrated to cover the real fabric strip behaviors. The proposed feedback-based folding was experimentally compared to two state-of-the-art folding methods and our method outperformed both of them in terms of accuracy.
We study the question of how to compute a point in the convex hull of an input set $S$ of $n$ points in ${mathbb R}^d$ in a differentially private manner. This question, which is trivial non-privately, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset $Gsubseteq{mathbb R}^d$, and furthermore, the size of $S$ must grow with the size of $G$. Previous works focused on understanding how $n$ needs to grow with $|G|$, and showed that $n=Oleft(d^{2.5}cdot8^{log^*|G|}right)$ suffices (so $n$ does not have to grow significantly with $|G|$). However, the available constructions exhibit running time at least $|G|^{d^2}$, where typically $|G|=X^d$ for some (large) discretization parameter $X$, so the running time is in fact $Omega(X^{d^3})$. In this paper we give a differentially private algorithm that runs in $O(n^d)$ time, assuming that $n=Omega(d^4log X)$. To get this result we study and exploit some structural properties of the Tukey levels (the regions $D_{ge k}$ consisting of points whose Tukey depth is at least $k$, for $k=0,1,...$). In particular, we derive lower bounds on their volumes for point sets $S$ in general position, and develop a rather subtle mechanism for handling point sets $S$ in degenerate position (where the deep Tukey regions have zero volume). A naive approach to the construction of the Tukey regions requires $n^{O(d^2)}$ time. To reduce the cost to $O(n^d)$, we use an approximation scheme for estimating the volumes of the Tukey regions (within their affine spans in case of degeneracy), and for sampling a point from such a region, a scheme that is based on the volume estimation framework of Lovasz and Vempala (FOCS 2003) and of Cousins and Vempala (STOC 2015). Making this framework differentially private raises a set of technical challenges that we address.
Planning accurate manipulation for deformable objects requires prediction of their state. The prediction is often complicated by a loss of stability that may result in collapse of the deformable object. In this work, stability of a fabric strip folding performed by a robot is studied. We show that there is a static instability in the folding process. This instability is detected in a physics-based simulation and the position of the instability is verified experimentally by real robotic manipulation. Three state-of-the-art methods for folding are assessed in the presence of static instability. It is shown that one of the existing folding paths is suitable for folding of materials with internal friction such as fabrics. Another folding path that utilizes dynamic motion exists for ideal elastic materials without internal friction. Our results show that instability needs to be considered in planning to obtain accurate manipulation of deformable objects.
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