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In this paper, we define a class of graph grammars that can be used to model and direct concurrent robotic self-assembly and similar self-organizing processes. We give several detailed examples of the formalism and then focus on the problem of synthesizing a grammar so that it generates a given, prespecified assembly. In particular, to generate an acyclic graph we synthesize a binary grammar (rules involve at most two parts), and for a general graph we synthesize a ternary grammar (rules involve at most three parts). In both cases, we characterize the number of concurrent steps required to achieve the assembly. We also show a general result that implies that no binary grammar can generate a unique stable assembly. We conclude the paper with a discussion of how graph grammars can be used to direct the self-assembly of robotic parts. [ABSTRACT FROM AUTHOR]

The problem of efficiently steering dynamical systems by generating input plans is considered. Plans are considered which consist of finite-length words constructed on an alphabet of input symbols, which could be, e.g., transmitted through a limited capacity channel to a remote system, where they can be decoded in suitable control actions. Efficiency is considered in terms of the computational complexity of plans, and in terms of their description length (in number of bits). We show that, by suitable choice of the control encoding, finite plans can be efficiently built for a wide class of dynamical systems, computing arbitrarily close approximations of a desired equilibrium in polynomial time. The paper also investigates how the efficiency of planning is affected by the choice of inputs, and provides some results as to optimal performance in terms of accuracy and range. [ABSTRACT FROM AUTHOR]

Combinatorial optimization over continuous and integer variables is a useful tool for solving complex optimal control problems of hybrid dynamical systems formulated in discrete- time. Current approaches are based on mixed-integer linear (or quadratic) programming (MIP), which provides the solution after solving a sequence of relaxed linear (or quadratic) programs. MIP formulations require the translation of the discrete/logic part of the hybrid problem into mixed-integer inequalities. Although this operation can be done automatically, most of the original symbolic structure of the problem (e.g., transition functions of finite state machines, logic constraints, symbolic variables, etc.) is lost during the conversion, with a consequent loss of computational performance. In this paper, we attempt to overcome such a difficulty by combining numerical techniques for solving convex programming problems with symbolic techniques for solving constraint satisfaction problems (CSP). The resulting ‘hybrid’ solver proposed here takes advantage of CSP solvers for dealing with satisfiability of logic constraints very efficiently. We propose a suitable model of the hybrid dynamics and a class of optimal control problems that embrace both symbolic and continuous variables/functions, and that are tailored to the use of the new hybrid solver. The superiority in terms of computational performance with respect to commercial MIP solvers is shown on a centralized supply chain management problem with uncertain forecast demand. [ABSTRACT FROM AUTHOR]

We discuss the generation of symbolic feedback con- trol sequences for navigating a sparsely-described and uncertain environment, together with the problem of sensing landmarks sufficiently well to make feedback meaningful. We explore the use of a symbolic control approach for mitigating the lack of a detailed map of the environment and for reducing the complexity associated with finding control laws which steer a control system between distant locations. Under our language-based approach, control inputs take the form of symbolic strings. The decision process that generates those strings is guided by estimates of the vehicle's location within a set of important landmarks and by the statistical effectiveness of each string. This arrangement, and in particular the symbolic nature of the control set, allows us to formulate and solve a class of optimal navigation problems which would be exceedingly difficult to handle if approached at the level of sensors and actuators. Our approach is illustrated in a series of numerical indoor navigation experiments. [ABSTRACT FROM AUTHOR]