Abstract—This paper describes a computational approach to designing a digital microfluidic system (DMFS) that can be rapidly reconfigured for new biochemical analyses. Such a “lab-on-a-chip” system for biochemical analysis, based on electrowetting or dielectrophoresis, must coordinate the motions of discrete droplets or biological cells using a planar array of electrodes. The authors have earlier introduced a layout-based system and demonstrated its flexibility through simulation, including the system’s ability to perform multiple assays simultaneously. Since array-layout design and droplet-routing strategies are closely related in such a DMFS, their goal is to provide designers with algorithms that enable rapid simulation and control of these DMFS devices. In this paper, the effects of variations in the basic array-layout design, droplet-routing control algorithms, and droplet spacing on system performance are characterized. DMFS arrays with hardware limited row-column addressing are considered, and a polynomial-time algorithm for coordinating droplet movement under such hardware limitations is developed. To demonstrate the capabilities of our system, we describe example scenarios, including dilution control and minimalist layouts, in which our system can be successfully applied. Index Terms—Array layout, biochips, digital microfluidics, droplet routing, lab-on-a-chip, performance analysis, row–column addressing. I.

In this paper we present an approach to coordinate the motions of droplets in digital microfluidic systems, a new class of lab-on-a-chip systems for biochemical analysis. A digital microfluidic system typically consists of a planar array of cells with electrodes that control the droplets. The primary challenge in using droplet-based systems is that they require the simultaneous coordination of a potentially large number of droplets on the array as the droplets move, mix, and split. In this paper we describe a general-purpose system that uses simple algorithms and yet is versatile. First, we present a semi-automated approach to generate the array layout in terms of components. Next, we discuss simple algorithms to select destination components for the droplets and a decentralized scheme for components to route the droplets on the array. These are then combined into a reconfigurable system that has been simulated in software to perform analyses such as the DNA polymerase chain reaction. The algorithms have been able to successfully coordinate hundreds of droplets simultaneously and perform one or more chemical analyses in parallel. Because it is challenging to analytically characterize the behavior of such systems, simulation methods to detect potential system instability are proposed.

We present an algorithm that computes the complete set of Pareto-optimal coordination strategies for two translating polygonal robots in the plane. A collision-free acyclic roadmap of piecewise-linear paths is given on which the two robots move. The robots have a maximum speed and are capable of instantly switching between any two arbitrary speeds. Each robot would like to minimize its travel time independently. The Pareto-optimal solutions are the ones for which there exist no solutions that are better for both robots. The algorithm computes exact solutions in time O(mn log n), in which m is the number of paths in the roadmap, n is the number of coordination space vertices. An implementation with computed examples is presented.

ABSTRACT. We consider coordination of multiple robots in a common environment, each robot having its own (distinct) roadmap. Our primary contribution is a classification of and exact algorithm for computing vector-valued — or Pareto — optima for collision-free coordination. We indicate the utility of new geometric techniques from CAT(0) geometry and give an argument that curvature bounds are the key distinguishing feature between systems for which the classification is finite and for those in which it is not. 1.

Abstract. Given a collection of robots sharing a common environment, assume that each possesses a graph (a 1-d complex also known as a roadmap) approximating its configuration space and, furthermore, that each robot wishes to travel to a goal while optimizing elapsed time. We consider vector-valued (or Pareto) optima for collision-free coordination on the product of these roadmaps with collision-type obstacles. Such optima are by no means unique: in fact, continua of Pareto optimal coordinations are possible. We prove a finite bound on the number of optimal coordinations in the physically relevant case where all obstacles are cylindrical (i.e., defined by pairwise collisions). The proofs rely crucially on perspectives from geometric group theory and cat(0) geometry. In particular, the finiteness bound depends on the fact that the associated coordination space is devoid of positive curvature. We also demonstrate that the finiteness bounds holds for systems with moving obstacles following known trajectories. 1. Introduction. 1.1. Motivation. In numerous settings, the coordination of multiple robots remains a basic and challenging research issue. Autonomous guided vehicles (AGVs) are used in a

In many multi-robot applications, the specific assignment of goal configurations to robots is less important than the overall behavior of the robot formation. In such cases, it is convenient to define a permutation-invariant multi-robot formation as a set of robot configurations, without assigning specific configurations to specific robots. For the case of robots that translate in the plane, we can represent such a formation by the coefficients of a complex polynomial whose roots represent the robot configurations. Since these coefficients are invariant with respect to permutation of the roots of the polynomial, they provide an effective representation for permutation-invariant formations. In this paper, we extend this idea to build a full representation of a permutation-invariant formation space. We describe the properties of the representation, and show how it can be used to construct collision-free paths for permutationinvariant formations.

This paper focuses on the coordination of multiple robots with kinodynamic constraints along specified paths. The presented approach generates continuous velocity profiles that avoid collisions and minimize the completion time for the robots. The approach identifies collision segments along each robot’s path and then optimizes the motions of the robots along their collision and collision-free segments. For each path segment for each robot, the minimum and maximum possible traversal times that satisfy the dynamics constraints are computed by solving the corresponding twopoint boundary value problems. Then the collision avoidance constraints for pairs of robots can be combined to formulate a mixed integer nonlinear programming (MINLP) problem. Since this nonconvex MINLP model is difficult to solve, we describe two related mixed integer linear programming (MILP) formulations that provide schedules that are lower and upper bounds on the optimum; the upper bound schedule is a continuous velocity schedule. The approach is illustrated with robots modeled as double integrators subject to velocity and acceleration constraints. An implementation that coordinates 12 nonholonomic car-like robots is described. 1

Abstract — In this paper, we study the following motion coordination problem: given n vehicles and n origin-destination pairs in the plane, what is the minimum time needed to transfer each vehicle from its origin to its destination, avoiding conflicts with other vehicles? The environment is free of obstacles and a conflict occurs when distance between any two vehicles is smaller than a velocity-dependent safety distance. In the case where the origin and destination points can be chosen arbitrarily, we show that the transfer takes Θ ( √ n ¯ L) time to complete, where ¯ L is the average distance between the origin and destination points. We also analyze the case in which origin and destination points are generated randomly according to a uniform distribution, and present an algorithm providing a constructive upper bound on the time needed to transfer vehicles from origins to their corresponding destination, proving that the transfer takes Θ ( √ n) time for this case. I.

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One of the fundamental tasks robots have to perform is planning their motions while avoiding collisions with obstacles in the environment. This is the central topic of the thesis. We restrict ourselves to motion planning for two- and three-dimensional rigid bodies and articulated robots moving in static and known virtual environments. This thesis has been divided into two parts. The first part deals with comparing and analyzing sampling-based motion planning techniques, in particular variants of the Probabilistic Roadmap Method (PRM). The PRM consists of two phases: a construction and a query phase. In the construction phase, a roadmap (graph) is built, approximating the motions that can be made in the environment. First, a free random sample is created. Such a sample describes a particular placement of the moving object (robot) in the workspace. Then, a simple local planner is employed to connect the sample to some neighbors. Samples and connections are added to the graph until the roadmap is dense enough. In the query phase, the start and goal samples are connected to the graph. The path is obtained by a Dijkstra's shortest path algorithm. Many variants of the PRM have been developed over the past decade. Using both time-based as well as reachability-based analysis, we compare some of the most prominent techniques. The results are surprising in the sense that techniques often perform differently than claimed by the designers. In addition, contrary to general belief, the main challenge is not getting the free space covered but getting the nodes connected, especially when the problems get more complicated, e.g. when a narrow passage is present. By using this knowledge, we can tackle the narrow passage problem by incorporating a more powerful local planner, a refined neighbor selection strategy and a hybrid sampling strategy. The analysis also shows why the PRM successfully deals with many motion planning problems. The second part deals with quality aspects of paths and roadmaps. A good path is relatively short, keeps some distance (clearance) from the obstacles, and is smooth. We will provide algorithms that increase path clearance. A big advantage of these algorithms is that high-clearance paths can now be efficiently created without using complex data structures and algorithms. We also elaborate on algorithms that successfully decrease path length. Then, we introduce the Reachability Roadmap Method which creates small roadmaps for two- and three-dimensional problems. Such a small roadmap has many advantages over a roadmap that is created by the PRM. In particular, the method assures low query times, low memory consumption, and the roadmap can be optimized easily. The algorithm also ensures that a path is always found (if one exists) at a given resolution. We unify the techniques to create high-quality roadmaps for interactive virtual environments. That is, we use the Reachability Roadmap Method to create an initial roadmap. We add useful cycles to provide alternative routes and short paths, and we add clearance to the roadmap to obtain high-clearance paths in real-time.