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In this paper, we introduce a robotic implementation of the theory of graph grammars (Klavins et al., 2005), which we use to model and direct self-organization in a formal, predictable and provably-correct fashion. The robots, which we call programmable parts, float passively on an air table and bind to each other upon random collisions. Once attached, they(More)
We describe how a graph grammar program for robotic self-assembly, together with measurements of kinetic rate data yield a Markov process model of the dynamics of programmed self-assembly. We demonstrate and validate the method by applying it to a physical testbed consisting of a number of "programmable parts", which are able to control their local(More)
— We consider the control of programmable self-assembling systems whose dynamics are governed by stochastic reaction-diffusion dynamics. In our system, particles may decide the outcomes of reactions initiated by the environment, thereby steering the global system to produce a desired assembly type. We describe a method that automatically generates a program(More)
We present derivative-based necessary and sufficient conditions ensuring player strategies constitute local Nash equilibria in non-cooperative continuous games. Our results can be interpreted as generalizations of analogous second-order conditions for local optimality from nonlinear programming and optimal control theory. Drawing on this analogy, we propose(More)
Rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs are ubiquitous simplifications introduced to reduce the complexity of mechanics models despite the obvious physical inaccuracies that each incurs individually. In concert, it is well known that the interaction of such idealized approximations can lead to conflicting and(More)
Legged robots are by nature strongly non-linear, high-dimensional systems whose full complexity permits neither tractable mathematical analysis nor comprehensive numerical study. In consequence, a growing body of literature interrogates simplified "template" (Full and Koditschek, 1999; Ghigliazza et al., 2005) models - to date almost exclusively confined to(More)
We present an integral feedback controller that regulates the average copy number of an assembly in a system of stochastically interacting robots. The mathematical model for these robots is a tunable reaction network, which makes this approach applicable to a large class of other systems, including ones that exhibit stochastic self-assembly at various(More)
We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous–time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constant– dimensional subsystem.(More)
— When the Poincaré map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates , we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in finite time. This result shows that the long-term behavior of a hybrid model with a large number of(More)
The study of controlled hybrid systems requires practical tools for approximation and comparison of system behaviors. Existing approaches to these problems impose undue restrictions on the system's continuous and discrete dynamics. Metrization and simulation of controlled hybrid systems is considered here in a unified framework by constructing a state space(More)