159 research outputs found
Searching Polyhedra by Rotating Half-Planes
The Searchlight Scheduling Problem was first studied in 2D polygons, where
the goal is for point guards in fixed positions to rotate searchlights to catch
an evasive intruder. Here the problem is extended to 3D polyhedra, with the
guards now boundary segments who rotate half-planes of illumination. After
carefully detailing the 3D model, several results are established. The first is
a nearly direct extension of the planar one-way sweep strategy using what we
call exhaustive guards, a generalization that succeeds despite there being no
well-defined notion in 3D of planar "clockwise rotation". Next follow two
results: every polyhedron with r>0 reflex edges can be searched by at most r^2
suitably placed guards, whereas just r guards suffice if the polyhedron is
orthogonal. (Minimizing the number of guards to search a given polyhedron is
easily seen to be NP-hard.) Finally we show that deciding whether a given set
of guards has a successful search schedule is strongly NP-hard, and that
deciding if a given target area is searchable at all is strongly PSPACE-hard,
even for orthogonal polyhedra. A number of peripheral results are proved en
route to these central theorems, and several open problems remain for future
work.Comment: 45 pages, 26 figure
Constrained Delaunay tetrahedral mesh generation and refinement
A {\it constrained Delaunay tetrahedralization} of a domain in is a tetrahedralization such that it respects the boundaries of this domain, and it has properties similar to those of a Delaunay tetrahedralization. Such objects have various applications such as finite element analysis, computer graphics rendering, geometric modeling, and shape analysis.
This article is devoted to presenting recent developments on constrained Delaunay tetrahedralizations of piecewise linear domains. The focus is for the application of numerically solving partial differential equations using finite element or finite volume methods. We survey various related results and detail two core algorithms that have provable guarantees and are amenable to practical implementation. We end this article by listing a set of open questions
Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments
The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades.
A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions.
The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated.
Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered
View-Invariant Regions and Mobile Robot Self-Localization
This paper addresses the problem of mobile robot self-localization
given a polygonal map and a set of observed edge segments. The
standard approach to this problem uses interpretation tree search with
pruning heuristics to match observed edges to map edges. Our approach
introduces a preprocessing step in which the map is decomposed into
'view-invariant regions' (VIRs). The VIR decomposition captures
information about map edge visibility, and can be used for a variety of
robot navigation tasks. Basing self-localization
search on VIRs greatly reduces the branching factor of the search
tree and thereby simplifies the search task. In this paper we define
the VIR decomposition and give algorithms for its computation and for
self-localization search. We present results of simulations comparing
standard and VIR-based search, and discuss the application of the VIR
decomposition to other problems in robot navigation
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
Planning in constraint space for multi-body manipulation tasks
Robots are inherently limited by physical constraints on their link lengths, motor torques, battery
power and structural rigidity. To thrive in circumstances that push these limits, such as in search
and rescue scenarios, intelligent agents can use the available objects in their environment as
tools. Reasoning about arbitrary objects and how they can be placed together to create useful
structures such as ramps, bridges or simple machines is critical to push beyond one's physical
limitations. Unfortunately, the solution space is combinatorial in the number of available objects
and the configuration space of the chosen objects and the robot that uses the structure is high
dimensional.
To address these challenges, we propose using constraint satisfaction as a means to test the
feasibility of candidate structures and adopt search algorithms in the classical planning literature
to find sufficient designs. The key idea is that the interactions between the components of a
structure can be encoded as equality and inequality constraints on the configuration spaces of the
respective objects. Furthermore, constraints that are induced by a broadly defined action, such as
placing an object on another, can be grouped together using logical representations such as Planning
Domain Definition Language (PDDL). Then, a classical planning search algorithm can reason about
which set of constraints to impose on the available objects, iteratively creating a structure that
satisfies the task goals and the robot constraints. To demonstrate the effectiveness of this
framework, we present both simulation and real robot results with static structures such as ramps,
bridges and stairs, and quasi-static structures such as lever-fulcrum simple machines.Ph.D
Énumération des rayons extrêmes d'un cône et applications en minimisation concave
Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal
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