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Register a new userEnumeration of Polyominoes & Polycubes Composed of Magnetic Cubes
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Added by
Aaron Becker
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper examines a family of designs for magnetic cubes and counts how many configurations are possible for each design as a function of the number of modules. Magnetic modular cubes are cubes with magnets arranged on their faces. The magnets are positioned so that each face has either magnetic south or north pole outward. Moreover, we require that the net magnetic moment of the cube passes through the center of opposing faces. These magnetic arrangements enable coupling when cube faces with opposite polarity are brought in close proximity and enable moving the cubes by controlling the orientation of a global magnetic field. This paper investigates the 2D and 3D shapes that can be constructed by magnetic modular cubes, and describes all possible magnet arrangements that obey these rules. We select ten magnetic arrangements and assign a colo to each of them for ease of visualization and reference. We provide a method to enumerate the number of unique polyominoes and polycubes that can be constructed from a given set of colored cubes. We use this method to enumerate all arrangements for up to 20 modules in 2D and 16 modules in 3D. We provide a motion planner for 2D assembly and through simulations compare which arrangements require fewer movements to generate and which arrangements are more common. Hardware demonstrations explore the self-assembly and disassembly of these modules in 2D and 3D.
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We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing faces of $Q$ to be covered multiple times. First, we define a variety of folding models according to whether the folds (a) must be along grid lines of $P$ or can divide squares in half (diagonally and/or orthogonally), (b) must be mountain or can be both mountain and valley, (c) can remain flat (forming an angle of $180^circ$), and (d) must lie on just the polycube surface or can have interior faces as well. Second, we give all the inclusion relations among all models that fold on the grid lines of $P$. Third, we characterize all polyominoes that can fold into a unit cube, in some models. Fourth, we give a linear-time dynamic programming algorithm to fold a tree-shaped polyomino into a constant-size polycube, in some models. Finally, we consider the triangular version of the problem, characterizing which polyiamonds fold into a regular tetrahedron.
Robotic materials are multi-robot systems formulated to leverage the low-order computation and actuation of the constituents to manipulate the high-order behavior of the entire material. We study the behaviors of ensembles composed of smart active particles, smarticles. Smarticles are small, low cost robots equipped with basic actuation and sensing abilities that are individually incapable of rotating or displacing. We demonstrate that a supersmarticle, composed of many smarticles constrained within a bounding membrane, can harness the internal collisions of the robotic material among the constituents and the membrane to achieve diffusive locomotion. The emergent diffusion can be directed by modulating the robotic material properties in response to a light source, analogous to biological phototaxis. The light source introduces asymmetries within the robotic material, resulting in modified populations of interaction modes and dynamics which ultimately result in supersmarticle biased locomotion. We present experimental methods and results for the robotic material which moves with a directed displacement in response to a light source.
The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that A) the average number of interior vertices over all convex polyominoes of perimeter $2n$ is asymptotic to $frac{n^2}{12}+frac{nsqrt{n}}{3sqrt{pi}} -frac{(21pi-16)n}{12pi}.$ B) the average number of boundary vertices with degree two over all convex polyominoes of perimeter $2n$ is asymptotic to $frac{n+6}{2}+frac{1}{sqrt{pi n}}+frac{(16-7pi)}{4pi n}.$ Additionally, we obtain an explicit generating function counting the number of convex polyominoes with $n$ boundary vertices of degrees at most three and show that this number is asymptotic to $ frac{n+1}{40}left(frac{3+sqrt{5}}{2}right)^{n-3} +frac{sqrt[4]{5}(2-sqrt{5})}{80sqrt{pi n}}left(frac{3+sqrt{5}}{2}right)^{n-2}. $ Moreover, we show that the expected number of the boundary vertices of degree four over all convex polyominoes with $n$ vertices of degrees at most three is asymptotically $ frac{n}{sqrt{5}}-frac{sqrt[4]{125}(sqrt{5}-1)sqrt{n}}{10sqrt{pi}}. $ C) the number of convex polyominoes with the outer-site perimeter $n$ is asymptotic to $frac{3(sqrt{5}-1)}{20sqrt{pi n}sqrt[4]{5}}left(frac{3+sqrt{5}}{2}right)^n,$ and show the expected number of the outer-site perimeter over all convex polyominoes with perimeter $2n$ is asymptotic to $frac{25n}{16}+frac{sqrt{n}}{4sqrt{pi}}+frac{1}{8}.$ Lastly, we prove that the expected perimeter over all convex polyominoes with the outer-site perimeter $n$ is asymptotic to $sqrt[4]{5}n$.
The paper focuses on the mechanics of a compliant serial manipulator composed of new type of dual-triangle elastic segments. Both the analytical and numerical methods were used to find the manipulator stable and unstable equilibrium configurations, as well as to predict corresponding manipulator shapes. The stiffness analysis was carried on for both loaded and unloaded modes, the stiffness matrices were computed using the Virtual Joint Method (VJM). The results demonstrate that either buckling or quasi-buckling phenomenon may occur under the loading, if the manipulator corresponding initial configuration is straight or non-straight one. Relevant simulation results are presented that confirm the theoretical study.
The paper focuses on the kinematics control of a compliant serial manipulator composed of a new type of dualtriangle elastic segments. Some useful optimization techniques were applied to solve the geometric redundancy problem, ensure the stability of the manipulator configurations with respect to the external forces/torques applied to the endeffector. The efficiency of the developed control algorisms is confirmed by simulation.
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