On reconfiguration of disks in the plane and related problems

We revisit two natural reconfiguration models for systems of disjoint objects in the plane: translation and sliding. Consider a set of n pairwise interior-disjoint objects in the plane that need to be brought from a given start (initial) configuration S into a desired goal (target) configuration T, without causing collisions. In the translation model, in one move an object is translated along a fixed direction to another position in the plane. In the sliding model, one move is sliding an object to another location in the plane by means of an arbitrarily complex continuous motion (that could involve rotations). We obtain various combinatorial and computational results for these two models: (I) For systems of n congruent disks in the translation model, Abellanas et al. showed that 2n − 1 moves always suffice and ⌊8n/5 ⌋ moves are sometimes necessary for transforming the start configuration into the target configuration. Here we further improve the lower bound to ⌊5n/3 ⌋ − 1, and thereby give a partial answer to one of their open problems. (II) We show that the reconfiguration problem with congruent disks in the translation model is NPhard, in both the labeled and unlabeled variants. This answers another open problem of Abellanas et al. (III) We also show that the reconfiguration problem with congruent disks in the sliding model is NP-hard, in both the labeled and unlabeled variants. (IV) For the reconfiguration with translations of n arbitrary convex bodies in the plane, 2n moves are always sufficient and sometimes necessary

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.

Motion Planning and Reconfiguration for Systems of Multiple Objects

This chapter surveys some recent results on motion planning and reconfiguration for systems of multiple objects and for modular systems with applications in robotics.

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Variational Methods and Planar Elliptic Growth

A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is proportional to the gradient of the domain's Green function with a fixed singularity on the interior. In this paper we review the Laplacian growth model and its key underlying assumptions, so that we may consider a generalization to so-called elliptic growth, wherein the Green function is replaced with that of a more general elliptic operator--this models, for example, inhomogeneities in the underlying plane. In this paper we continue the development of the underlying mathematics for elliptic growth, considering perturbations of the Green function due to those of the driving operator, deriving characterizations and examples of growth, developing a weak formulation of growth via balayage, and discussing of a couple of inverse problems in the spirit of Calder\'on. We conclude with a derivation of a more delicate, reregularized model for Hele-Shaw flow

Motion planning and reconfiguration for systems of multiple objects

Abstract This chapter surveys some recent results on motion planning and reconfiguration for systems of multiple objects and for modular systems with applications in robotics

SIMBA: Tendon-Driven Modular Continuum Arm with Soft Reconfigurable Gripper

In this paper, we describe the conceptual design and implementation of the Soft Compliant Manipulator for Broad Applications (SIMBA) manipulator, which is designed and developed for participating in the RoboSoft Grand Challenge 2016. In our novel design, we have proposed (1) a modular continuum arm with independent actuation units for each module, to increase maintainability; (2) a soft reconfigurable hand, for a better adaptation of the fingers to objects of different shapes and size; (3) a moving base for increasing the workspace. We used a hybrid approach in designing and manufacturing by integrating soft and hard components, in both materials and actuation, providing high lateral stiffness in the arm through flat springs, soft joints in fingers for more compliancy and tendon-motor actuation mechanism providing flexibility but at the same time precision and speed. The SIMBA manipulator has demonstrated excellent grasping and manipulation capabilities by being able to grasp objects with different fragility, geometry, and size; and by lifting objects with up to 2 kg of weight it demonstrate also to be robust and reliable. The experimental results pointed out that our design and approach can lead to the realization of robots able to act in unknown and unstructured environments in synergy with humans, for a variety of applications where compliancy is fundamental, preserving robustness and safety

Controlled mobility in stochastic and dynamic wireless networks

We consider the use of controlled mobility in wireless networks where messages arriving randomly in time and space are collected by mobile receivers (collectors). The collectors are responsible for receiving these messages via wireless transmission by dynamically adjusting their position in the network. Our goal is to utilize a combination of wireless transmission and controlled mobility to improve the throughput and delay performance in such networks. First, we consider a system with a single collector. We show that the necessary and sufficient stability condition for such a system is given by ρ<1 where ρ is the expected system load. We derive lower bounds for the expected message waiting time in the system and develop policies that are stable for all loads ρ<1 and have asymptotically optimal delay scaling. We show that the combination of mobility and wireless transmission results in a delay scaling of Θ([1 over 1−ρ]) with the system load ρ, in contrast to the Θ([1 over (1−ρ)[superscript 2]]) delay scaling in the corresponding system without wireless transmission, where the collector visits each message location. Next, we consider the system with multiple collectors. In the case where simultaneous transmissions to different collectors do not interfere with each other, we show that both the stability condition and the delay scaling extend from the single collector case. In the case where simultaneous transmissions to different collectors interfere with each other, we characterize the stability region of the system and show that a frame-based version of the well-known Max-Weight policy stabilizes the system asymptotically in the frame length.National Science Foundation (U.S.) (Grant CNS-0915988)United States. Army Research Office. Multidisciplinary University Research Initiative (Grant W911NF-08-1-0238