Dustin Reishus

Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature

We consider the problem of fault-tolerance in nanoscale algorithmic self-assembly. We employ a standard variant of Winfree’s abstract Tile Assembly Model (aTAM), the two-handed aTAM, in which square “tiles” – a model of molecules constructed from DNA for the purpose of engineering self-assembled nanostructures – aggregate according to specific binding sites of varying strengths, and in which large aggregations of tiles may attach to each other, in contrast to the seeded aTAM, in which tiles aggregate one at a time to a single specially designated “seed” assembly. We focus on a major cause of errors in tile-based self-assembly: that of unintended growth due to “weak” strength-1 bonds, which if allowed to persist, may be stabilized by subsequent attachment of neighboring tiles in the sense that at least energy 2 is now required to break apart the resulting assembly, i.e., the errant assembly is stable at temperature 2. We study a common self-assembly benchmark problem, that of assembling an n×n square using O(log n) unique tile types, under the two-handed model of self-assembly. Our main result achieves a much stronger notion of fault-tolerance than those achieved previously. Arbitrary strength-1 growth is allowed, however, any assembly that grows sufficiently to become stable at temperature 2 is guaranteed to assemble into the correct final assembly of an n×n square. In other words, errors due to insufficient attachment, which is the cause of errors studied in earlier papers on fault-tolerance, are prevented absolutely in our main construction, rather than only with high probability and for sufficiently small structures, as in previous fault tolerance studies.

The Undecidability of the Infinite Ribbon Problem: Implications for Computing by Self-Assembly

Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Recent experiments in self-assembly demonstrate its potential for the parallel creation of a large number of nanostructures, including possibly computers. A systematic study of self-assembly as a mathematical process has been initiated by L. Adleman and E. Winfree. The individual components are modeled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific “glue,” and two adjacent tiles will stick iff they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional “structures” such as squares and rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called a ribbon: a non-self-crossing rectilinear sequence of tiles on the plane, in which successive tiles are adjacent along an edge and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. While the problem can be proved undecidable using existing techniques if the ribbon is required to start with a given “seed” tile, our result settles the “unseeded” case, an open problem formerly known as the “unlimited infinite snake problem.” The proof is based on a construction, due to R. Robinson, of a special set of tiles that allow only aperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the “strong plane-filling property”—a variation of the “plane-filling property” previously defined by J. Kari. A construction of “sandwich” tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable tiling problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A “motif” construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable. An immediate consequence of our result is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set.

On the Mathematics of the Law of Mass Action

In 1864, Waage and Guldberg formulated the “law of mass action.” Since that time, chemists, chemical engineers, physicists and mathematicians have amassed a great deal of knowledge on the topic. In our view, sufficient understanding has been acquired to warrant a formal mathematical consolidation. A major goal of this consolidation is to solidify the mathematical foundations of mass action chemistry—to provide precise definitions, elucidate what can now be proved, and indicate what is only conjectured. In addition, we believe that the law of mass action is of intrinsic mathematical interest and should be made available in a form that might transcend its application to chemistry alone. We present the law of mass action in the context of a dynamical theory of sets of binomials over the complex numbers.

Miniature six-channel range and bearing system: Algorithm, analysis and experimental validation.

We present an algorithm, analysis, and implementation of a six-channel range and bearing system for swarm robot systems with sizes in the order of centimeters. The proposed approach relies on a custom sensor and receiver model, and collection of intensity signals from all possible sensor/emitter pairs. This allows us to improve range calculation by accounting for orientation-dependent variations in the transmitted intensity, as well as to determine the orientation of the emitting robot. We show how the algorithm and analysis generalize to other range and bearing systems, and evaluate its performance experimentally using two ping-pong ball-sized “Droplets” mounted on a precise gantry system.

Path finding in the tile assembly model

Swarm robotics, active self-assembly, and amorphous computing are fields that focus on designing systems of large numbers of small, simple components that can cooperate to complete complex tasks. Many of these systems are inspired by biological systems, and all attempt to use the simplest components and environments possible, while still being capable of achieving their goals. The canonical problems for such biologically-inspired systems are shape assembly and path finding. In this paper, we demonstrate path finding in the well-studied tile assembly model, a model of molecular self-assembly that is strictly simpler than other biologically-inspired models. As in related work, our systems function in the presence of obstacles and can be made fault-tolerant. The path-finding systems use @Q(1) distinct components and find minimal-length paths in time linear in the length of the path.

Object Transportation by Granular Convection Using Swarm Robots

We propose a novel method for object transport using granular convection, in which the granular material is a robot swarm consisting of small robots with minimal sensors. Granular convection is commonly observed in the “Brazil Nut Effect”. In this work, we consider the transported object to be passive, however, and not actuated like the surrounding granular material. We show that the passive object can be transported to a given destination in spite of the fact that each robot does not know the location of the object being transported nor the location of the destination. Each robot moves based solely on a weak repulsive force from the destination and stochastic perturbations. We first show fundamental characteristics of a system with no communication between robots. We observe that very high or very low robot densities are detrimental to object transport. We then show that heterogeneous swarms increase performance. We propose two types of heterogeneous swarm systems: a swarm in which robots switch states probabilistically, and a swarm in which state propagates using local communication. The signal propagation system shows the best performance in terms of success rate and accuracy in a wide range of densities.

Precise truss assembly using commodity parts and low precision welding

Hardware and software design and system integration for an intelligent precision jigging robot (IPJR), which allows high precision assembly using commodity parts and low-precision bonding, is described. Preliminary 2D experiments that are motivated by the problem of assembling space telescope optical benches and very large manipulators on orbit using inexpensive, stock hardware and low-precision welding are also described. An IPJR is a robot that acts as the precise “jigging”, holding parts of a local structure assembly site in place, while an external low precision assembly agent cuts and welds members. The prototype presented in this paper allows an assembly agent (for this prototype, a human using only low precision tools), to assemble a 2D truss made of wooden dowels to a precision on the order of millimeters over a span on the order of meters. The analysis of the assembly error and the results of building a square structure and a ring structure are discussed. Options for future work, to extend the IPJR paradigm to building in 3D structures at micron precision are also summarized.

A stick-slip omnidirectional powertrain for low-cost swarm robotics: Mechanism, calibration, and control

We present an omnidirectional powertrain for swarm robotic platforms that relies on low-cost vibration motors.We describe a mechanism and controller to achieve full 3-DoF motion on the plane. The proposed approach does not require the motors to be in phase, and overcomes differences in manufacturing by a hardware-in-the-loop auto-calibration routine based on the Nelder-Mead algorithm, which issues motion commands via infrared and records the resulting trajectories using an off-the-shelf webcam. We show convergence results of the calibration routine and sample trajectories of the swarm robotic platform “Droplet” demonstrating turning and omnidirectional drive.

Connecting the Dots: Molecular Machinery for Distributed Robotics

Nature is considered one promising area to search for inspiration in designing robotic systems. Some work in swarm robotics has tried to build systems that resemble distributed biological systems and inherit biologys fault tolerance, scalability, dependability, and robustness. Such systems, as well as ones in the areas of active self-assembly and amorphous computing, typically use relatively simple components with limited computation, memory, and computational power to accomplish complex tasks, such as forming paths in the presence of obstacles. We demonstrate that such tasks can be accomplished in the well-studied tile assembly model, a model of molecular self-assembly that is strictly simpler than other biologically-inspired models. Our systems use a small number of distinct components to find minimal-length paths in time linear in the length of the path while inheriting scalability and fault tolerance of the underlying natural process of self-assembly.