Magnetic modular millibots, capable of deterministic self-assembly and reconfiguration under wireless magnetic fields, offer a promising route toward mesoscale manipulation in structured and confined environments. This work presents a modular chain millibot composed of cubic units with free-to-spin internal magnets that align under external fields. While individual cubes cannot propel independently, their assembly into chains enables controlled locomotion through sliding and tumbling modes. We first demonstrate open-loop operation in confined workspaces, including chain formation, navigation, controlled disassembly, and wall climbing, supported by a dynamic model and parametric analysis that identify the actuation and geometric conditions required for successful climbing. Building on this foundation, we introduce a closed-loop control framework that integrates vision-based feedback. As the millibot’s movement speed increases with chain length, the order of assembly strongly affects task completion time; we therefore formulate the sequence-planning problem as a harmonic traveling salesman problem (HTSP) and solve it to compute deterministic cube-collection sequences that minimize the effective travel cost. The controller applies sliding and tumbling dynamics to realize obstacle-aware navigation and reliable self-assembly. Experiments validate autonomous assembly of four cubes with a two-cube chain into a six-cube chain in free space and collection of three cubes with wall climbing in a confined workspace, both with 100% success. The measured mean unit travel times were 0.67 s/mm in free space and 0.89 s/mm in confined environments. Collectively, these results establish a robust automation framework for reversible mesoscale self-assembly, programmable navigation, and lab-on-chip applications.

Abstract When considering motion planning for a swarm of n labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, collision-free moves. The objective is to reach the new configuration in a minimum amount of time. Problems of this type have been considered before, with recent notable results achieving constant stretch for parallel reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, the total duration of an overall schedule can be bounded to $$\mathcal {O}(d)$$ O ( d ) , which is optimal up to constant factors. An important constraint for coordinated reconfiguration is to keep the swarm connected after each time step. In previous work, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations of unlabeled robots; on the other hand, the existence of non-constant lower bounds on the stretch factor was unknown. We resolve these major open problems by (1) establishing a lower bound of $$\Omega (\sqrt{n})$$ Ω ( n ) for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected, labeled reconfiguration can be achieved. In addition, we show that (3) it is -complete to decide whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a schedule of makespan 1 exists.

We investigate motion planning algorithms for the assembly of shapes in the tilt model in which unit-square tiles move in a grid world under the influence of uniform external forces and self-assemble according to certain rules. We provide several heuristics and experimental evaluation of their success rate, solution length, and runtime.

When considering motion planning for a swarm of $n$ labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, collision-free robot motions. The objective is to reach the new configuration in a minimum amount of time; an important constraint is to keep the swarm connected at all times. Problems of this type have been considered before, with recent notable results achieving constant stretch for not necessarily connected reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of $d$, the total duration of an overall schedule can be bounded to $\mathcal{O}(d)$, which is optimal up to constant factors. However, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations (which arise by increasing all dimensions of a given object by the same multiplicative factor) of unlabeled robots. We resolve these major open problems by (1) establishing a lower bound of $Ω(\sqrt{n})$ for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected reconfiguration can be achieved. In addition, we show that (3) it is NP-complete to decide whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.

Abstract We provide algorithmic methods for connected reconfiguration of lattice-based cellular structures by finite-state robots, motivated by large-scale constructions in space. We present algorithms that are able to detect and reconfigure arbitrary polyominoes, while also preserving connectivity of a structure during reconfiguration; we also provide mathematical proofs and performance guarantees. Specific results include methods for determining a bounding box, scaling a given arrangement, and adapting more general algorithms for transforming polyominoes.

Abstract In micro- and nano-scale systems, particles can be moved by using an external force like gravity or a magnetic field. In the presence of adhesive particles that can attach to each other, the challenge is to decide whether a shape is constructible. Previous work provides a class of shapes for which constructibility can be decided efficiently when particles move maximally into the same direction induced by a global signal. In this paper we consider the single step model, i.e., a model in which each particle moves one unit step into the given direction. We restrict the assembly process such that at each single time step actually one particle is added to and moved within the workspace. We prove that deciding constructibility is NP-complete for three-dimensional shapes, and that a maximum constructible shape can be approximated. The same approximation algorithm applies for 2D. We further present linear-time algorithms to decide whether or not a tree-shape in 2D or 3D is constructible. Scaling a shape yields constructibility; in particular we show that the 2-scaled copy of every non-degenerate polyomino is constructible. In the three-dimensional setting we show that the 3-scaled copy of every non-degenerate polycube is constructible.

We illustrate and animate the classic problem of deciding whether a given graph has an Eulerian path. Starting with a collection of instances of increasing difficulty, we present a set of pictorial instructions, and show how they can be used to solve all instances. These IDEA instructions ("A series of nonverbal algorithm assembly instructions") have proven to be both entertaining for experts and enlightening for novices. We (w)rap up with a song and dance to Euler’s original instance.

In this video, we consider recognition and reconfiguration of lattice-based cellular structures by very simple robots with only basic functionality. The underlying motivation is the construction and modification of space facilities of enormous dimensions, where the combination of new materials with extremely simple robots promises structures of previously unthinkable size and flexibility. We present algorithmic methods that are able to detect and reconfigure arbitrary polyominoes, based on finite-state robots, while also preserving connectivity of a structure during reconfiguration. Specific results include methods for determining a bounding box, scaling a given arrangement, and adapting more general algorithms for transforming polyominoes.

In this video, we present theoretical and practical methods for achieving arbitrary reconfiguration of a set of objects, based on the use of external forces, such as a magnetic field or gravity: Upon actuation, each object is pushed in the same direction. This concept can be used for a wide range of applications in which particles do not have their own energy supply or in which they are subject to the same global control commands. A crucial challenge for achieving any desired target configuration is breaking global symmetry in a controlled fashion. Previous work (some of which was presented during SoCG 2015) made use of specifically placed barriers; however, introducing precisely located obstacles into the workspace is impractical for many scenarios. In this paper, we present a different, less intrusive method: making use of the interplay between static friction with a boundary and the external force to achieve arbitrary reconfiguration. Our key contributions are theoretical characterizations of the critical coefficient of friction that is sufficient for rearranging two particles in triangles, convex polygons, and regular polygons; a method for reconfiguring multiple particles in rectangular workspaces, and deriving practical algorithms for these rearrangements. Hardware experiments show the efficacy of these procedures, demonstrating the usefulness of this novel approach.

We investigate algorithmic approaches for targeted drug delivery in a complex, maze-like environment, such as a vascular system. The basic scenario is given by a large swarm of micro-scale particles ("agents") and a particular target region ("tumor") within a system of passageways. Agents are too small to contain on-board power or computation and are instead controlled by a global external force that acts uniformly on all particles, such as an applied fluidic flow or electromagnetic field. The challenge is to deliver all agents to the target region with a minimum number of actuation steps. We provide a number of results for this challenge. We show that the underlying problem is NP-hard, which explains why previous work did not provide provably efficient algorithms. We also develop a number of algorithmic approaches that greatly improve the worst-case guarantees for the number of required actuation steps. We evaluate our algorithmic approaches by a number of simulations, both for deterministic algorithms and searches supported by deep learning, which show that the performance is practically promising.

We consider recognition and reconfiguration of lattice-based cellular structures by very simple robots with only basic functionality. The underlying motivation is the construction and modification of space facilities of enormous dimensions, where the combination of new materials with extremely simple robots promises structures of previously unthinkable size and flexibility; this is also closely related to the newly emerging field of programmable matter. Aiming for large-scale scalability, both in terms of the number of the cellular components of a structure, as well as the number of robots that are being deployed for construction requires simple yet robust robots and mechanisms, while also dealing with various basic constraints, such as connectivity of a structure during reconfiguration. To this end, we propose an approach that combines ultra-light, cellular building materials with extremely simple robots. We develop basic algorithmic methods that are able to detect and reconfigure arbitrary cellular structures, based on robots that have only constant-sized memory. As a proof of concept, we demonstrate the feasibility of this approach for specific cellular materials and robots that have been developed at NASA.

We present algorithmic results for the parallel assembly of many micro-scale objects in two and three dimensions from tiny particles, which has been proposed in the context of programmable matter and self-assembly for building high-yield micro-factories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle. Particles bond when forced together with another appropriate particle. Due to the physical and geometric constraints, not all shapes can be built in this manner; this gives rise to the Tilt Assembly Problem (TAP) of deciding constructibility. For simply-connected polyominoes P in 2D consisting of N unit-squares (“tiles”), we prove that TAP can be decided in \(O(N\log N)\) time. For the optimization variant MaxTAP (in which the objective is to construct a subshape of maximum possible size), we show polyAPX -hardness: unless P = NP , MaxTAP cannot be approximated within a factor of \(\Omega (N^{\frac{1}{3}})\) ; for tree-shaped structures, we give an \(\Omega (N^{\frac{1}{2}})\) -approximation algorithm. For the efficiency of the assembly process itself, we show that any constructible shape allows pipelined assembly, which produces copies of P in O (1) amortized time, i.e., N copies of P in O ( N ) time steps. These considerations can be extended to three-dimensional objects: For the class of polycubes P we prove that it is NP -hard to decide whether it is possible to construct a path between two points of P ; it is also NP -hard to decide constructibility of a polycube P . Moreover, it is expAPX -hard to maximize a sequentially constructible path from a given start point.

Abstract We study parallel online algorithms: For some fixed integer k, a collective of k parallel processes that perform online decisions on the same sequence of events forms a k-copy algorithm. For any given time and input sequence, the overall performance is determined by the best of the k individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with advice on future events, i.e., a number of bits available in advance. Parallel online algorithms are also of interest in practical scenarios in which redundancy is used for hedging against undesired outcomes. We develop Predictive Harmonic $$_3$$ 3 (PH3), a relatively simple family of k-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing k. In particular, we show that $$k=6$$ k = 6 suffices to guarantee a factor of 1.5714 for PH3, which is better than 1.57829, the performance of the best known 1-copy algorithm Advanced Harmonic, while $$k=11$$ k = 11 suffices to achieve a factor of 1.5406, beating the known lower bound of 1.54278 for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of 1.54278, which is considerably less than the previous bound of 15 bits.

This paper studies the general problem of physically sorting polyominoes according to shape using a 2D, rigid, grid-based workspace. The workspace is designed for sensorless operation, using a fixed set of open-loop force-field inputs that move a polyomino from an inlet port to an outlet port that corresponds to the polyomino's shape, and reset the workspace to classify the next polyomino. This paper proves that static workspaces can classify all orthoconvex polyominoes of width w and height h, and provides a motion sequence and required size of workspace as a function of wand h. By allowing moving polyomino cams that assist in the sorting, we can design dynamic works paces that can sort all polyomi-noes that are “completely filled” using a constant number of force-field inputs. Hardware experiments using magnetic and gravity-based actuation demonstrate these static and dynamic sensorless classifiers at the millimeter scale.

Abstract We contribute results for a set of fundamental problems in the context of programmable matter by presenting algorithmic methods for evaluating and manipulating a collective of particles by a finite automaton that can neither store significant amounts of data, nor perform complex computations, and is limited to a handful of possible physical operations. We provide a toolbox for carrying out fundamental tasks on a given arrangement of particles, using the arrangement itself as a storage device, similar to a higher-dimensional Turing machine with geometric properties. Specific results include time- and space-efficient procedures for bounding, counting, copying, reflecting, rotating or scaling a complex given shape.

In 1991, Edelsbrunner and Tan gave an O(n2) algorithm for finding the MinMax Length triangulation of a set of points in the plane, but stated the complexity of finding a MaxMin Edge Length Triangulation (MELT) as a natural open problem. We resolve this long-standing problem by showing that computing a MELT is NP-complete. Moreover, we prove that (unless P=NP), there is no polynomial-time approximation algorithm that can approximate MELT within any polynomial factor. While this may be taken as conclusive evidence from a theoretical point of view that the problem is hopelessly intractable, it still makes sense to consider powerful optimization methods, such as integer programming (IP), in order to obtain provably optimal solutions for intances of non-trivial size. A straightforward IP based on pairwise disjointness of the Θ(n2) segments between the n points has Θ(n4) constraints, making this IP hopelessly intractable from a practical point of view, even for relatively small n. The main algorithm engineering twist of this paper is to demonstrate how the combination of geometric insights with refined methods of combinatorial optimization can still help to put together an exact method capable of computing optimal MELT solutions for planar point sets up to n = 200. Our key idea is to exploit specific geometric properties in combination with more compact IP formulations, such that we are able to drastically reduce the IPs. On the practical side, we combine two of the most powerful software packages for the individual components: CGAL for carrying out the geometric computations, and CPLEX for solving the IPs. In addition, we discuss specific analytic aspects of the speedup for random point sets.