Arslan, Omur
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Publication Discriminative Measures for Comparison of Phylogenetic Trees(2017-01-01) Arslan, Omur; Guralnik, Dan P.; Koditschek, Daniel E.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*LabPublication Sensor-Based Reactive Navigation in Unknown Convex Sphere Worlds(2018-05-28) Arslan, Omur; Koditschek, Daniel E.We construct a sensor-based feedback law that provably solves the real-time collision-free robot navigation problem in a compact convex Euclidean subset cluttered with unknown but sufficiently separated and strongly convex obstacles. Our algorithm introduces a novel use of separating hyperplanes for identifying the robot’s local obstacle-free convex neighborhood, affording a reactive (online-computed) continuous and piecewise smooth closed-loop vector field whose smooth flow brings almost all configurations in the robot’s free space to a designated goal location, with the guarantee of no collisions along the way. Specialized attention to planar navigable environments yields a necessary and sufficient condition on convex obstacles for almost global navigation towards any goal location in the environment. We further extend these provable properties of the planar setting to practically motivated limited range, isotropic and anisotropic sensing models, and the nonholonomically constrained kinematics of the standard differential drive vehicle. We conclude with numerical and experimental evidence demonstrating the effectiveness of the proposed sensory feedback motion planner.Publication A Recursive, Distributed Minimum Spanning Tree Algorithm for Mobile Ad Hoc Networks(2014-07-01) Arslan, Omur; Koditschek, Daniel EWe introduce a recursive (“anytime”) distributed algorithm that iteratively restructures any initial spanning tree of a weighted graph towards a minimum spanning tree while guaranteeing at each successive step a spanning tree shared by all nodes that is of lower weight than the previous. Each recursive step is computed by a different active node at a computational cost at most quadratic in the total number of nodes and at a communications cost incurred by subsequent broadcast of the new edge set over the new spanning tree. We show that a polynomial cubic in the number of nodes bounds the worst case number of such steps required to reach a minimum spanning tree and, hence, the number of broadcasts along the way. We conjecture that the distributed, anytime nature of this algorithm is particularly suited to tracking minimum spanning trees in (sufficiently slowly changing) mobile ad hoc networks.Publication On the Optimality of Napoleon Triangles(2016-03-09) Arslan, Omur; Koditschek, Daniel E.An elementary geometric construction, known as Napoleon’s theorem, produces an equilateral triangle, obtained from equilateral triangles erected on the sides of any initial triangle: The centers of the three equilateral triangles erected on the sides of the arbitrarily given original triangle, all outward or all inward, are the vertices of the new equilateral triangle. In this note, we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle, whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle, whose vertices are closest to the original in the sense of minimizing the sum of the three squared distances. For more information: Kod*labPublication Coordinated Robot Navigation via Hierarchical Clustering(2016-03-21) Arslan, Omur; Guralnik, Dan P.; Koditschek, Daniel E.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. For more information: Kod*labPublication Integration of Local Geometry and Metric Information in Sampling-Based Motion Planning(2018-02-25) Pacelli, Vincent; Arslan, Omur; Koditschek, Daniel E.The efficiency of sampling-based motion planning algorithms is dependent on how well a steering procedure is capable of capturing both system dynamics and configuration space geometry to connect sample configurations. This paper considers how metrics describing local system dynamics may be combined with convex subsets of the free space to describe the local behavior of a steering function for sampling-based planners. Subsequently, a framework for using these subsets to extend the steering procedure to incorporate this information is introduced. To demonstrate our framework, three specific metrics are considered: the LQR cost-to-go function, a Gram matrix derived from system linearization, and the Mahalanobis distance of a linear-Gaussian system. Finally, numerical tests are conducted for a second-order linear system, a kinematic unicycle, and a linear-Gaussian system to demonstrate that our framework increases the connectivity of sampling-based planners and allows them to better explore the free space. For more information: Kod*lab.Publication Smooth Extensions of Feedback Motion Planners via Reference Governors(2017-05-29) Arslan, Omur; Koditschek, Daniel E.In robotics, it is often practically and theoretically convenient to design motion planners for approximate low-order (e.g., position- or velocity-controlled) robot models first, and then adapt such reference planners to more accurate high-order (e.g., force/torque-controlled) robot models. In this paper, we introduce a novel provably correct approach to extend the applicability of low-order feedback motion planners to high-order robot models, while retaining stability and collision avoidance properties, as well as enforcing additional constraints that are specific to the high-order models. Our smooth extension framework leverages the idea of reference governors to separate the issues of stability and constraint satisfaction, affording a bidirectionally coupled robot-governor system where the robot ensures stability with respect to the governor and the governor enforces state (e.g., collision avoidance) and control (e.g., actuator limits) constraints. We demonstrate example applications of our framework for augmenting path planners and vector field planners to the second-order robot dynamics.Publication Anytime Hierarchical Clustering(2014-04-13) Arslan, Omur; Koditschek, Daniel EWe propose a new anytime hierarchical clustering method that iteratively transforms an arbitrary initial hierarchy on the configuration of measurements along a sequence of trees we prove for a fixed data set must terminate in a chain of nested partitions that satisfies a natural homogeneity requirement. Each recursive step re-edits the tree so as to improve a local measure of cluster homogeneity that is compatible with a number of commonly used (e.g., single, average, complete) linkage functions. As an alternative to the standard batch algorithms, we present numerical evidence to suggest that appropriate adaptations of this method can yield decentralized, scalable algorithms suitable for distributed/parallel computation of clustering hierarchies and online tracking of clustering trees applicable to large, dynamically changing databases and anomaly detection.Publication Exact Robot Navigation Using Power Diagrams(2016-05-16) Arslan, Omur; Koditschek, Daniel E.We reconsider the problem of reactive navigation in sphere worlds, i.e., the construction of a vector field over a compact, convex Euclidean subset punctured by Euclidean disks, whose flow brings a Euclidean disk robot from all but a zero measure set of initial conditions to a designated point destination, with the guarantee of no collisions along the way. We use power diagrams, generalized Voronoi diagrams with additive weights, to identify the robot’s collision free convex neighborhood, and to generate the value of our proposed candidate solution vector field at any free configuration via evaluation of an associated convex optimization problem. We prove that this scheme generates a continuous flow with the specified properties. We also propose its practical extension to the nonholonomically constrained kinematics of the standard differential drive vehicle. For more information: https://kodlab.seas.upenn.eduPublication Sensory Steering for Sampling-Based Motion Planning(2017-01-01) Arslan, Omur; Pacelli, Vincent; Koditschek, Daniel E.Sampling-based algorithms offer computationally efficient, practical solutions to the path finding problem in high-dimensional complex configuration spaces by approximately capturing the connectivity of the underlying space through a (dense) collection of sample configurations joined by simple local planners. In this paper, we address a long-standing bottleneck associated with the difficulty of finding paths through narrow passages. Whereas most prior work considers the narrow passage problem as a sampling issue (and the literature abounds with heuristic sampling strategies) very little attention has been paid to the design of new effective local planners. Here, we propose a novel sensory steering algorithm for sampling- based motion planning that can “feel” a configuration space locally and significantly improve the path planning performance near difficult regions such as narrow passages. We provide computational evidence for the effectiveness of the proposed local planner through a variety of simulations which suggest that our proposed sensory steering algorithm outperforms the standard straight-line planner by significantly increasing the connectivity of random motion planning graphs. For more information: Kod*lab