Coordinated Robot Navigation via Hierarchical Clustering

We introduce the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. Traditionally an unsupervised learning method, hierarchical clustering offers a formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions by relating the continuous space of configurations to the combinatorial space of trees. We formalize and exploit this relation, developing computationally effective reactive algorithms for navigating through the combinatorial space in concert with geometric realizations for a particular choice of hierarchical clustering method. These constructions yield computationally effective vector field planners for both hierarchically invariant as well as transitional navigation in the configuration space. We apply these methods to the centralized coordination and control of $n$ perfectly sensed and actuated Euclidean spheres in a $d$-dimensional ambient space (for arbitrary $n$ and $d$). Given a desired configuration supporting a desired hierarchy, we construct a hybrid controller which is quadratic in $n$ and algebraic in $d$ and prove that its execution brings all but a measure zero set of initial configurations to the desired goal with the guarantee of no collisions along the way.Comment: 29 pages, 13 figures, 8 tables, extended version of a paper in preparation for submission to a journa

Toward a Memory Model for Autonomous Topological Mapping and Navigation: the Case of Binary Sensors and Discrete Actions

We propose a self-organizing database for per- ceptual experience capable of supporting autonomous goal- directed planning. The main contributions are: (i) a formal demonstration that the database is complex enough in principle to represent the homotopy type of the sensed environment; (ii) some initial steps toward a formal demonstration that the database offers a computationally effective, contractible approximation suitable for motion planning that can be ac- cumulated purely from autonomous sensory experience. The provable properties of an effectively trained data-base exploit certain notions of convexity that have been recently generalized for application to a symbolic (discrete) representation of subset nesting relations. We conclude by introducing a learning scheme that we conjecture (but cannot yet prove) will be capable of achieving the required training, assuming a rich enough exposure to the environment. For more information: Kod*La

Universal Memory Architectures for Autonomous Machines

We propose a self-organizing memory architecture (UMA) for perceptual experience provably capable of supporting autonomous learning and goal-directed problem solving in the absence of any prior information about the agent’s environment. The architecture is simple enough to ensure (1) a quadratic bound (in the number of available sensors) on space requirements, and (2) a quadratic bound on the time-complexity of the update-execute cycle. At the same time, it is sufficiently complex to provide the agent with an internal representation which is (3) minimal among all representations which account for every sensory equivalence class consistent with the agent’s belief state; (4) capable, in principle, of recovering a topological model of the problem space; and (5) learnable with arbitrary precision through a random application of the available actions. These provable properties — both the trainability and the operational efficacy of an effectively trained memory structure — exploit a duality between weak poc sets — a symbolic (discrete) representation of subset nesting relations — and non-positively curved cubical complexes, whose rich convexity theory underlies the planning cycle of the proposed architecture

Iterated Belief Revision Under Resource Constraints: Logic as Geometry

We propose a variant of iterated belief revision designed for settings with limited computational resources, such as mobile autonomous robots. The proposed memory architecture---called the universal memory architecture (UMA)---maintains an epistemic state in the form of a system of default rules similar to those studied by Pearl and by Goldszmidt and Pearl (systems Z and Z+). A duality between the category of UMA representations and the category of the corresponding model spaces, extending the Sageev-Roller duality between discrete poc sets and discrete median algebras provides a two-way dictionary from inference to geometry, leading to immense savings in computation, at a cost in the quality of representation that can be quantified in terms of topological invariants. Moreover, the same framework naturally enables comparisons between different model spaces, making it possible to analyze the deficiencies of one model space in comparison to others. This paper develops the formalism underlying UMA, analyzes the complexity of maintenance and inference operations in UMA, and presents some learning guarantees for different UMA-based learners. Finally, we present simulation results to illustrate the viability of the approach, and close with a discussion of the strengths, weaknesses, and potential development of UMA-based learners

Clustering-Based Robot Navigation and Control

In robotics, it is essential to model and understand the topologies of configuration spaces in order to design provably correct motion planners. The common practice in motion planning for modelling configuration spaces requires either a global, explicit representation of a configuration space in terms of standard geometric and topological models, or an asymptotically dense collection of sample configurations connected by simple paths. In this short note, we present an overview of our recent results that utilize clustering for closing the gap between these two complementary approaches. Traditionally an unsupervised learning method, clustering offers automated tools to discover hidden intrinsic structures in generally complex-shaped and high-dimensional configuration spaces of robotic systems. We demonstrate some potential applications of such clustering tools to the problem of feedback motion planning and control. In particular, we briefly present our use of hierarchical clustering for provably correct, computationally efficient coordinated multirobot motion design, and we briefly describe how robot-centric Voronoi diagrams can be used for provably correct safe robot navigation in forest-like cluttered environments, and for provably correct collision-free coverage and congestion control of heterogeneous disk-shaped robots.For more information: Kod*la

Discriminative Measures for Comparison of Phylogenetic Trees

In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a “combing” of the Nearest Neighbor Interchange (NNI) graph of binary hierarchies, providing an efficient approximation to the (NP-hard) NNI distance in terms of “edit length”. At the same time, a closed form formula for $d_{nav}$ presents it as a weighted count of pairwise incompatibilities between clusters, lending it the character of an edge dissimilarity measure as well. A relaxation of this formula to a simple count yields another measure on all trees — the crossing dissimilarity $d_{CM}$. Both dissimilarities are symmetric and positive definite (vanish only between identical trees) on binary hierarchies but they fail to satisfy the triangle inequality. Nevertheless, both are bounded below by the widely used Robinson–Foulds metric and bounded above by a closely related true metric, the cluster-cardinality metric $d_{CC}$. We show that each of the three proposed new dissimilarities is computable in time O($n^2$) in the number of leaves $n$, and conclude the paper with a brief numerical exploration of the distribution over tree space of these dissimilarities in comparison with the Robinson–Foulds metric and the more recently introduced matching-split distance. For more information: Kod*La