Computing Shortest Paths in the Plane with Removable Obstacles

We consider the problem of computing a Euclidean shortest path in the presence of removable obstacles in the plane. In particular, we have a collection of pairwise-disjoint polygonal obstacles, each of which may be removed at some cost c_i > 0. Given a cost budget C > 0, and a pair of points s, t, which obstacles should be removed to minimize the path length from s to t in the remaining workspace? We show that this problem is NP-hard even if the obstacles are vertical line segments. Our main result is a fully-polynomial time approximation scheme (FPTAS) for the case of convex polygons. Specifically, we compute an (1 + epsilon)-approximate shortest path in time O({nh}/{epsilon^2} log n log n/epsilon) with removal cost at most (1+epsilon)C, where h is the number of obstacles, n is the total number of obstacle vertices, and epsilon in (0, 1) is a user-specified parameter. Our approximation scheme also solves a shortest path problem for a stochastic model of obstacles, where each obstacle\u27s presence is an independent event with a known probability. Finally, we also present a data structure that can answer s-t path queries in polylogarithmic time, for any pair of points s, t in the plane

Revisiting the Minimum Constraint Removal Problem in Mobile Robotics

The minimum constraint removal problem seeks to find the minimum number of constraints, i.e., obstacles, that need to be removed to connect a start to a goal location with a collision-free path. This problem is NP-hard and has been studied in robotics, wireless sensing, and computational geometry. This work contributes to the existing literature by presenting and discussing two results. The first result shows that the minimum constraint removal is NP-hard for simply connected obstacles where each obstacle intersects a constant number of other obstacles. The second result demonstrates that for $n$ simply connected obstacles in the plane, instances of the minimum constraint removal problem with minimum removable obstacles lower than $(n+1)/3$ can be solved in polynomial time. This result is also empirically validated using several instances of randomly sampled axis-parallel rectangles.Comment: Accepted for presentation at the 18th international conference on Intelligent Autonomous System 202

Memory-Constrained Algorithms for Simple Polygons

A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of $O(\log n)$ bits, where $n$ is the size of the input. We assume that a simple $n$-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in $O(n^2)$ time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow $s$ words of working space. After a preprocessing of $O(n^2)$ time, we are able to solve shortest path queries between any two points inside the polygon in $O(n^2/s)$ time.Comment: Preprint appeared in EuroCG 201

Approximation Algorithms for Geometric Networks

The main contribution of this thesis is approximation algorithms for several computational geometry problems. The underlying structure for most of the problems studied is a geometric network. A geometric network is, in its abstract form, a set of vertices, pairwise connected with an edge, such that the weight of this connecting edge is the Euclidean distance between the pair of points connected. Such a network may be used to represent a multitude of real-life structures, such as, for example, a set of cities connected with roads. Considering the case that a specific network is given, we study three separate problems. In the first problem we consider the case of interconnected `islands' of well-connected networks, in which shortest paths are computed. In the second problem the input network is a triangulation. We efficiently simplify this triangulation using edge contractions. Finally, we consider individual movement trajectories representing, for example, wild animals where we compute leadership individuals. Next, we consider the case that only a set of vertices is given, and the aim is to actually construct a network. We consider two such problems. In the first one we compute a partition of the vertices into several subsets where, considering the minimum spanning tree (MST) for each subset, we aim to minimize the largest MST. The other problem is to construct a $t$-spanner of low weight fast and simple. We do this by first extending the so-called gap theorem. In addition to the above geometric network problems we also study a problem where we aim to place a set of different sized rectangles, such that the area of their corresponding bounding box is minimized, and such that a grid may be placed over the rectangles. The grid should not intersect any rectangle, and each cell of the grid should contain at most one rectangle. All studied problems are such that they do not easily allow computation of optimal solutions in a feasible time. Instead we consider approximation algorithms, where near-optimal solutions are produced in polynomial time. In addition to the above geometric network problems we also study a problem where we aim to place a set of different sized rectangles, such that the area of their corresponding bounding box is minimized, and such that a grid may be placed over the rectangles. The grid should not intersect any rectangle, and each cell of the grid should contain at most one rectangle. All studied problems are such that they do not easily allow computation of optimal solutions in a feasible time. Instead we consider approximation algorithms, where near-optimal solutions are produced in polynomial time

Improved Approximation Bounds for the Minimum Constraint Removal Problem

In the minimum constraint removal problem, we are given a set of geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable, and (perhaps surprisingly) no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is a new approximation technique that gives O(sqrt{n})-approximation for rectangles, disks as well as rectilinear polygons. The technique also gives O(sqrt{n})-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem

Robust geometric forest routing with tunable load balancing

Although geometric routing is proposed as a memory-efficient alternative to traditional lookup-based routing and forwarding algorithms, it still lacks: i) adequate mechanisms to trade stretch against load balancing, and ii) robustness to cope with network topology change. The main contribution of this paper involves the proposal of a family of routing schemes, called Forest Routing. These are based on the principles of geometric routing, adding flexibility in its load balancing characteristics. This is achieved by using an aggregation of greedy embeddings along with a configurable distance function. Incorporating link load information in the forwarding layer enables load balancing behavior while still attaining low path stretch. In addition, the proposed schemes are validated regarding their resilience towards network failures

Smoothing of Piecewise Linear Paths

We present an anytime-capable fast deterministic greedy algorithm for smoothing piecewise linear paths consisting of connected linear segments. With this method, path points with only a small influence on path geometry (i.e. aligned or nearly aligned points) are successively removed. Due to the removal of less important path points, the computational and memory requirements of the paths are reduced and traversing the path is accelerated. Our algorithm can be used in many different applications, e.g. sweeping, path finding, programming-by-demonstration in a virtual environment, or 6D CNC milling. The algorithm handles points with positional and orientational coordinates of arbitrary dimension

Design of a strategy to obtain safe paths from collaborative robot teamwork

Documento en PDF a color.figuras, tablasThis doctoral thesis was designed and implemented using a strategy of explorer agents and a management and monitoring system to obtain the shortest and safest paths. The strategy was simulated using Matlab R2016 in 10 test environments. The comparisons were made between the results obtained by considering each robot's work and contrasting it with the results obtained by implementing the cooperative-collaborative strategy. For this purpose, were used two path planning algorithms, they are the A* and the Greedy Best First Search (GBFS). Some changes were made to these classic algorithms to improve their performance to guarantee interactions and comparisons between them, transforming them into Incremental Heuristic (IH) algorithms, which gave rise to a couple of agents with new path planners called IH-A* and IH-GBFS. The cooperative strategy was implemented with IH-A* and IH-GBFS algorithms to obtain the shortest paths. The cooperative process was used 300 times in 100 complete tests (3 times in 10 tests in each of 10 environments), which allowed determining that the strategy decreased the original path (without cooperation) in 79% of the cases. In 20.50% of cases, the author identified that the cooperative process, reduced to less than half the original path. The collaborative strategy was implemented to obtain the safer path, using a communications system that allows the interaction among the explorer agents, the test environment, and the management and monitoring system to generate early warnings and compare the risk between paths. In this work, the risk is due to hidden marks found by the explorer agents; for this reason, it is implemented a potential risk function that allows obtaining the path risk estimated. The path risk estimated metric is the one that facilitates the evaluation and comparison of risk between paths to find safer paths. The AWMRs operates using a kinematic model, a controller, a path planner, and sensors that allow them to navigate through the environment gently and safely. Simultaneously with the explorer agents, the administration and monitoring system as a user interface that facilitates the presentation and consolidation of results were implemented. Subsequently, 16 tests were carried out, implementing the complete cooperative-collaborative strategy in four different environments, which had hidden marks. When analyzing the results, it was determined that the Shortest Safest Estimated Path was found in 62.5% of the tests. A WMR and a square test stage were built. In the test scenario, 240 path tracking tests were carried out (the WMR travelled 24 different paths; the WMR travelled each path ten times). The path data were obtained using odometry with encoders onboard the robot and image processing through an external camera. The author apply a tracking error analysis on the WMR path, travelling a circumference of 3.64 m in length. When comparing the path obtained with the WMR kinematic model with the data obtained using image processing, a Mean Absolute Percentage Error (MAPE) of 2,807% was obtained; and with the odometry data, the MAPE was 1,224%. As a general conclusion, this study has numerically identified the relevance of the implementation of the cooperative-collaborative strategy in robotic teamwork to find shortest and safest paths, a strategy applied in test environments that have obstacles and hidden marks. The cooperative-collaborative strategy can be used in different applications that involve displacement in a dangerous place or environment, such as a minefield or a region at risk of spreading COVID-19.Esta tesis doctoral fue diseñada e implementada utilizando una estrategia de agentes exploradores y un sistema de gestión y seguimiento para obtener caminos más cortos y seguros. La estrategia se simuló utilizando Matlab R2016 en 10 entornos de prueba. Las comparaciones se realizaron entre los resultados obtenidos al considerar el trabajo realizado por cada robot y contrastarlo con los resultados obtenidos al implementar la estrategia cooperativa-colaborativa. Para ello, se utilizaron dos algoritmos de planificación de rutas, que son el A* y el Greedy Best First Search (GBFS). Se realizaron algunos cambios a estos algoritmos clásicos para mejorar su rendimiento para garantizar interacciones y comparaciones entre ellos, transformándolos en algoritmos Heurísticos Incrementales (IH), lo que dio lugar a un par de agentes con nuevos planificadores de rutas denominados IH-A * e IH- GBFS. La estrategia cooperativa se implementó con algoritmos IH-A * e IH-GBFS para obtener los caminos más cortos. El proceso cooperativo se utilizó 300 veces en 100 pruebas completas (3 veces en 10 pruebas en cada uno de los 10 entornos), lo que permitió determinar que la estrategia disminuyó la trayectoria original (sin cooperación) en el 79% de los casos. En el 20,50% de los casos, el autor identificó que el proceso cooperativo, redujo la distancia entre inicio y meta a menos de la mitad del recorrido original. La estrategia colaborativa se implementó para obtener el camino más seguro, utilizando un sistema de comunicaciones que permite la interacción entre los agentes exploradores, el entorno de prueba y el sistema de gestión y monitoreo para generar alertas tempranas y comparar el riesgo entre caminos. En este trabajo, el riesgo se debe a las marcas ocultas encontradas por los agentes exploradores; por ello, se implementa una función de riesgo potencial que permite obtener el riesgo de ruta estimado. La métrica estimada de riesgo de ruta es la que facilita la evaluación y comparación de riesgo entre rutas para encontrar rutas más seguras. Los robots autónomos móviles con ruedas (en inglés AWMR) operan utilizando un modelo cinemático, un controlador, un planificador de rutas y sensores que les permiten navegar por el entorno de manera suave y segura. Simultáneamente con los agentes exploradores, el autor implementó un sistema de administración y monitoreo como interfaz de usuario que facilita la presentación y consolidación de resultados. Posteriormente, se realizaron 16 pruebas, implementando la estrategia cooperativa-colaborativa completa en cuatro entornos diferentes, que tenían marcas ocultas. Al analizar los resultados, se determinó que una ruta estimada más corta y más segura se obtenía en el 62.5% de las pruebas. Se construyeron un WMR y un escenario de prueba cuadrado. En el escenario de prueba, se llevaron a cabo 240 pruebas de seguimiento de ruta (el WMR recorrió 24 rutas diferentes; el WMR recorrió cada ruta diez veces). Los datos de la trayectoria se obtuvieron utilizando odometría con encoders a bordo del robot y procesamiento de imágenes a través de una cámara externa. El autor aplica un análisis de error de seguimiento en la ruta recorrida por el WMR, generando una circunferencia de 3,64 m de longitud. Al comparar la ruta obtenida con el modelo cinemático del WMR con los datos obtenidos usando el procesamiento de imágenesse obtuvo un error de porcentaje absoluto medio (MAPE) de 2.807%; y con los datos de odometría, el MAPE fue de 1,224%. Como conclusión general, este estudio ha identificado numéricamente la relevancia de la implementación de la estrategia cooperativa-colaborativa en el trabajo en equipo robótico para encontrar caminos más cortos y seguros, estrategia aplicada en entornos de prueba que poseen obstáculos y marcas ocultas. La estrategia cooperativa-colaborativa puede ser utilizada en diferentes aplicaciones que involucran el desplazamiento en un lugar o entorno peligroso, como pueden ser un campo minado o una región en riesgo de propagación de COVID-19.DoctoradoDoctor en Ingeniería - Ingeniería Automátic

Axel Rover Tethered Dynamics and Motion Planning on Extreme Planetary Terrain

Some of the most appealing science targets for future exploration missions in our solar system lie in terrains that are inaccessible to state-of-the-art robotic rovers such as NASA's Opportunity, thereby precluding in situ analysis of these rich opportunities. Examples of potential high-yield science areas on Mars include young gullies on sloped terrains, exposed layers of bedrock in the Victoria Crater, sources of methane gas near Martian volcanic ranges, and stepped delta formations in heavily cratered regions. In addition, a recently discovered cryovolcano on Titan and frozen water near the south pole of our own Moon could provide a wealth of knowledge to any robotic explorer capable of accessing these regions. To address the challenge of extreme terrain exploration, this dissertation presents the Axel rover, a two-wheeled tethered robot capable of rappelling down steep slopes and traversing rocky terrain. Axel is part of a family of reconfigurable rovers, which, when docked, form a four-wheeled vehicle nicknamed DuAxel. DuAxel provides untethered mobility to regions of extreme terrain and serves as an anchor support for a single Axel when it undocks and rappels into low-ground. Axel's performance on extreme terrain is primarily governed by three key system components: wheel design, tether control, and intelligent planning around obstacles. Investigations in wheel design and optimizing for extreme terrain resulted in the development of grouser wheels. Experiments demonstrated that these grouser wheels were very effective at surmounting obstacles, climbing rocks up to 90% of the wheel diameter. Terramechanics models supported by experiments showed that these wheels would not sink excessively or become trapped in deformable terrain. Predicting tether forces in different configurations is also essential to the rover's mobility. Providing power, communication, and mobility forces, the tether is Axel's lifeline while it rappels steep slopes, and a cut, abraded, or ruptured tether would result in an untimely end to the rover's mission. Understanding tether forces are therefore paramount, and this thesis both models and measures tension forces to predict and avoid high-stress scenarios. Finally, incorporating autonomy into Axel is a unique challenge due to the complications that arise during tether management. Without intelligent planning, rappelling systems can easily become entangled around obstacles and suffer catastrophic failures. This motivates the development of a novel tethered planning algorithm, presented in this thesis, which is unique for rappelling systems. Recent field experiments in natural extreme terrains on Earth demonstrate the Axel rover's potential as a candidate for future space operations. Both DuAxel and its rappelling counterpart are rigorously tested on a 20 meter escarpment and in the Arizona desert. Through analysis and experiments, this thesis provides the framework for a new generation of robotic explorers capable of accessing extreme planetary regions and potentially providing clues for life beyond Earth.</p