22 research outputs found
Distributed boundary tracking using alpha and Delaunay-Cech shapes
For a given point set in a plane, we develop a distributed algorithm to
compute the shape of . shapes are well known geometric
objects which generalize the idea of a convex hull, and provide a good
definition for the shape of . We assume that the distances between pairs of
points which are closer than a certain distance are provided, and we show
constructively that this information is sufficient to compute the alpha shapes
for a range of parameters, where the range depends on .
Such distributed algorithms are very useful in domains such as sensor
networks, where each point represents a sensing node, the location of which is
not necessarily known.
We also introduce a new geometric object called the Delaunay-\v{C}ech shape,
which is geometrically more appropriate than an shape for some cases,
and show that it is topologically equivalent to shapes
A fractal dimension for measures via persistent homology
We use persistent homology in order to define a family of fractal dimensions,
denoted for each homological dimension
, assigned to a probability measure on a metric space. The case
of -dimensional homology () relates to work by Michael J Steele (1988)
studying the total length of a minimal spanning tree on a random sampling of
points. Indeed, if is supported on a compact subset of Euclidean space
for , then Steele's work implies that
if the absolutely continuous part of
has positive mass, and otherwise .
Experiments suggest that similar results may be true for higher-dimensional
homology , though this is an open question. Our fractal dimension is
defined by considering a limit, as the number of points goes to infinity,
of the total sum of the -dimensional persistent homology interval lengths
for random points selected from in an i.i.d. fashion. To some
measures we are able to assign a finer invariant, a curve measuring the
limiting distribution of persistent homology interval lengths as the number of
points goes to infinity. We prove this limiting curve exists in the case of
-dimensional homology when is the uniform distribution over the unit
interval, and conjecture that it exists when is the rescaled probability
measure for a compact set in Euclidean space with positive Lebesgue measure
Near-Optimal Motion Planning Algorithms Via A Topological and Geometric Perspective
Motion planning is a fundamental problem in robotics, which involves finding a path for an autonomous system, such as a robot, from a given source to a destination while avoiding collisions with obstacles. The properties of the planning space heavily influence the performance of existing motion planning algorithms, which can pose significant challenges in handling complex regions, such as narrow passages or cluttered environments, even for simple objects. The problem of motion planning becomes deterministic if the details of the space are fully known, which is often difficult to achieve in constantly changing environments. Sampling-based algorithms are widely used among motion planning paradigms because they capture the topology of space into a roadmap. These planners have successfully solved high-dimensional planning problems with a probabilistic-complete guarantee, i.e., it guarantees to find a path if one exists as the number of vertices goes to infinity. Despite their progress, these methods have failed to optimize the sub-region information of the environment for reuse by other planners. This results in re-planning overhead at each execution, affecting the performance complexity for computation time and memory space usage.
In this research, we address the problem by focusing on the theoretical foundation of the algorithmic approach that leverages the strengths of sampling-based motion planners and the Topological Data Analysis methods to extract intricate properties of the environment. The work contributes a novel algorithm to overcome the performance shortcomings of existing motion planners by capturing and preserving the essential topological and geometric features to generate a homotopy-equivalent roadmap of the environment. This roadmap provides a mathematically rich representation of the environment, including an approximate measure of the collision-free space. In addition, the roadmap graph vertices sampled close to the obstacles exhibit advantages when navigating through narrow passages and cluttered environments, making obstacle-avoidance path planning significantly more efficient.
The application of the proposed algorithms solves motion planning problems, such as sub-optimal planning, diverse path planning, and fault-tolerant planning, by demonstrating the improvement in computational performance and path quality. Furthermore, we explore the potential of these algorithms in solving computational biology problems, particularly in finding optimal binding positions for protein-ligand or protein-protein interactions.
Overall, our work contributes a new way to classify routes in higher dimensional space and shows promising results for high-dimensional robots, such as articulated linkage robots. The findings of this research provide a comprehensive solution to motion planning problems and offer a new perspective on solving computational biology problems
Computational intelligence approaches to robotics, automation, and control [Volume guest editors]
No abstract available
Policy space abstraction for a lifelong learning agent
This thesis is concerned with policy space abstractions that concisely encode alternative
ways of making decisions; dealing with discovery, learning, adaptation and use of these
abstractions. This work is motivated by the problem faced by autonomous agents that
operate within a domain for long periods of time, hence having to learn to solve many
different task instances that share some structural attributes. An example of such a
domain is an autonomous robot in a dynamic domestic environment. Such environments
raise the need for transfer of knowledge, so as to eliminate the need for long learning
trials after deployment.
Typically, these tasks would be modelled as sequential decision making problems,
including path optimisation for navigation tasks, or Markov Decision Process models for
more general tasks. Learning within such models often takes the form of online learning
or reinforcement learning. However, handling issues such as knowledge transfer and
multiple task instances requires notions of structure and hierarchy, and that raises several
questions that form the topic of this thesis – (a) can an agent acquire such hierarchies in
policies in an online, incremental manner, (b) can we devise mathematically rigorous
ways to abstract policies based on qualitative attributes, (c) when it is inconvenient to
employ prolonged trial and error learning, can we devise alternate algorithmic methods
for decision making in a lifelong setting?
The first contribution of this thesis is an algorithmic method for incrementally
acquiring hierarchical policies. Working with the framework of options - temporally
extended actions - in reinforcement learning, we present a method for discovering
persistent subtasks that define useful options for a particular domain. Our algorithm
builds on a probabilistic mixture model in state space to define a generalised and
persistent form of ‘bottlenecks’, and suggests suitable policy fragments to make options.
In order to continuously update this hierarchy, we devise an incremental process which
runs in the background and takes care of proposing and forgetting options. We evaluate
this framework in simulated worlds, including the RoboCup 2D simulation league
domain.
The second contribution of this thesis is in defining abstractions in terms of equivalence
classes of trajectories. Utilising recently developed techniques from computational
topology, in particular the concept of persistent homology, we show that a library of
feasible trajectories could be retracted to representative paths that may be sufficient for
reasoning about plans at the abstract level. We present a complete framework, starting
from a novel construction of a simplicial complex that describes higher-order connectivity
properties of a spatial domain, to methods for computing the homology of this
complex at varying resolutions. The resulting abstractions are motion primitives that
may be used as topological options, contributing a novel criterion for option discovery.
This is validated by experiments in simulated 2D robot navigation, and in manipulation
using a physical robot platform.
Finally, we develop techniques for solving a family of related, but different, problem
instances through policy reuse of a finite policy library acquired over the agent’s lifetime.
This represents an alternative approach when traditional methods such as hierarchical
reinforcement learning are not computationally feasible. We abstract the policy space
using a non-parametric model of performance of policies in multiple task instances, so
that decision making is posed as a Bayesian choice regarding what to reuse. This is
one approach to transfer learning that is motivated by the needs of practical long-lived
systems. We show the merits of such Bayesian policy reuse in simulated real-time
interactive systems, including online personalisation and surveillance
Computational intelligence approaches to robotics, automation, and control [Volume guest editors]
No abstract available
NASA SBIR abstracts of 1991 phase 1 projects
The objectives of 301 projects placed under contract by the Small Business Innovation Research (SBIR) program of the National Aeronautics and Space Administration (NASA) are described. These projects were selected competitively from among proposals submitted to NASA in response to the 1991 SBIR Program Solicitation. The basic document consists of edited, non-proprietary abstracts of the winning proposals submitted by small businesses. The abstracts are presented under the 15 technical topics within which Phase 1 proposals were solicited. Each project was assigned a sequential identifying number from 001 to 301, in order of its appearance in the body of the report. Appendixes to provide additional information about the SBIR program and permit cross-reference of the 1991 Phase 1 projects by company name, location by state, principal investigator, NASA Field Center responsible for management of each project, and NASA contract number are included
Aeronautical engineering: A continuing bibliography with indexes (supplement 301)
This bibliography lists 1291 reports, articles, and other documents introduced into the NASA scientific and technical information system in Feb. 1994. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics
Persistence spectral sequences
This Doctoral thesis is centered on connections between persistent homology and
spectral sequences. We explain some of the approaches in the literature exploring this
connection. Our main focus is on Mayer-Vietoris spectral sequences associated to filtered covers on filtered complexes. A particular case of this spectral sequence is used
for measuring exact changes on barcode decompositions under small perturbations
of the underlying data. On the other hand, these objects allow for a setup to parallelize persistent homology computations, while retaining useful information related
to the chosen covers. We explore some generalizations of the traditional setup to diagrams of regular complexes consisting of regular morphisms; these become useful for
working with non-sparse complexes. In addition, we explore stability results related
to these new invariants, both with respect to local changes and with respect to changes
on the chosen covering sets. Finally, we present some computational experiments by
the use of PERMAVISS which illustrate some of these ideas