Constrained Shortest Paths in Terrains and Graphs

Finding a shortest path is one of the most well-studied optimization problems. In this thesis we focus on shortest paths in geometric and graph theoretic settings subject to different feasibility constraints that arise in practical applications of such paths. One of the most fundamental problems in computational geometry is finding shortest paths in terrains, which has many applications in robotics, computer graphics and Geographic Information Systems (GISs). There are many variants of the problem in which the feasibility of a path is determined by some geometric property of the terrain. One such variant is the shortest descending path (SDP) problem, where the feasible paths are those that always go downhill. We need to compute an SDP, for example, for laying a canal of minimum length from the source of water at the top of a mountain to fields for irrigation purpose, and for skiing down a mountain along a shortest route. The complexity of finding SDPs is open. We give a full characterization of the bend angles of an SDP, showing that they follow a generalized form of Snell's law of refraction of light. We also reduce the SDP problem to the problem of finding an SDP through a given sequence of faces, by adapting the sequence tree approach of Chen and Han for our problem. Our results have two implications. First, we isolate the difficult aspect of SDPs. The difficulty is not in deciding which face sequence to use, but in finding the SDP through a given face sequence. Secondly, our results help us identify some classes of terrains for which the SDP problem is solvable in polynomial time. We give algorithms for two such classes. The difficulty of finding an exact SDP motivates the study of approximation algorithms for the problem. We devise two approximation algorithms for SDPs in general terrains---these are the first two algorithms to handle the SDP problem in such terrains. The algorithms are robust and easy-to-implement. We also give two approximation algorithms for the case when a face sequence is given. The first one solves the problem by formulating it as a convex optimization problem. The second one uses binary search together with our characterization of the bend angles of an SDP to locate an approximate path. We introduce a generalization of the SDP problem, called the shortest gently descending path (SGDP) problem, where a path descends but not too steeply. The additional constraint to disallow a very steep descent makes the paths more realistic in practice. For example, a vehicle cannot follow a too steep descent---this is why a mountain road has hairpin bends. We give two easy-to-implement approximation algorithms for SGDPs, both using the Steiner point approach. Between a pair of points there can be many SGDPs with different number of bends. In practice an SGDP with fewer bends or smaller total turn-angle is preferred. We show using a reduction from 3-SAT that finding an SGDP with a limited number of bends or a limited total turn-angle is hard. The hardness result applies to a generalization of the SGDP problem called the shortest anisotropic path problem, which is a well-studied computational geometry problem with many practical applications (e.g., robot motion planning), yet of unknown complexity. Besides geometric shortest paths, we also study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an easy-to-implement algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The algorithm handles a forbidden path using vertex replication, i.e., replicating vertices and judiciously deleting edges so as to remove the forbidden path but keep all of its subpaths. The main challenge is that vertex replication can result in an exponential number of copies of any forbidden path that overlaps the current one. The algorithm couples vertex replication with the "growth" of a shortest path tree in such a way that the extra copies of forbidden paths produced during vertex replication are immaterial

Motion planning on steep terrain for the tethered axel rover

This paper considers the motion planning problem that arises when a tethered robot descends and ascends steep obstacle-strewn terrain. This work is motivated by the Axel tethered robotic rover designed to provide access to extreme extra-planetary terrains. Motion planning for this type of rover is very different from traditional planning problems because the tether geometry under high loading must be considered during the planning process. Furthermore, only round-trip paths that avoid tether entanglement are viable solutions to the problem. We present an algorithm for tethered robot motion planning on steep terrain that reduces the likelihood that the tether will become entangled during descent and ascent of steep slopes. The algorithm builds upon the notion of the shortest homotopic tether path and its associated sleeve. We provide a simple example for purposes of illustration

Flow Computations on Imprecise Terrains

We study the computation of the flow of water on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time

Tethered Motion Planning for a Rappelling Robot

The Jet Propulsion Laboratory and Caltech developed the Axel rover to investigate and demonstrate the potential for tethered extreme terrain mobility, such as allowing access to science targets on the steep crater walls of other planets. Tether management is a key issue for Axel and other rappelling rovers. Avoiding tether entanglement constrains the robot's valid motions to the set of outgoing and returning path pairs that are homotopic to each other. In the case of a robot on a steep slope, a motion planner must additionally ensure that this ascent-descent path pair is feasible, based on the climbing forces provided by the tether. This feasibility check relies on the taut tether configuration, which is the shortest path in the homotopy class of the ascent-descent path pair. This dissertation presents a novel algorithm for tethered motion planning in extreme terrains, produced by combining shortest-homotopic-path algorithms from the topology and computational geometry communities with traditional graph search methods. The resulting tethered motion planning algorithm searches for this shortest path, checks for feasibility, and then generates waypoints for an ascent-descent path pair in the same homotopy class. I demonstrate the implementation of this algorithm on a Martian crater data set such as might be seen for a typical mission. By searching only for the shortest path, and ordering that search according to a heuristic, this algorithm proceeds more efficiently than previous tethered path-planning algorithms for extreme terrain. Frictional tether-terrain interaction may cause dangerously intermittent and unstable tether obstacles, which can be categorized based on their stability. Force-balance equations from the rope physics literature provide a set of tether and terrain conditions for static equilibrium, which can be used to determine if a given tether configuration will stick to a given surface based on tether tension. By estimating the tension of Axel's tether when driving, I divide potential tether tension obstacles into the following categories: acting as obstacles, acting as non-obstacles, and hazardous intermittent obstacles where it is uncertain whether the tether would slip or stick under normal driving tension variance. This dissertation describes how to modify the obstacle map as the categorization of obstacles fluctuates, and how to alter a motion plan around the dangerous tether friction obstacles. Together, these algorithms and methods form a framework for tethered motion planning on extreme terrain.</p

MAP: Medial Axis Based Geometric Routing in Sensor Networks

One of the challenging tasks in the deployment of dense wireless networks (like sensor networks) is in devising a routing scheme for node to node communication. Important consideration includes scalability, routing complexity, the length of the communication paths and the load sharing of the routes. In this paper, we show that a compact and expressive abstraction of network connectivity by the medial axis enables efficient and localized routing. We propose MAP, a Medial Axis based naming and routing Protocol that does not require locations, makes routing decisions locally, and achieves good load balancing. In its preprocessing phase, MAP constructs the medial axis of the sensor field, defined as the set of nodes with at least two closest boundary nodes. The medial axis of the network captures both the complex geometry and non-trivial topology of the sensor field. It can be represented compactly by a graph whose size is comparable with the complexity of the geometric features (e.g., the number of holes). Each node is then given a name related to its position with respect to the medial axis. The routing scheme is derived through local decisions based on the names of the source and destination nodes and guarantees delivery with reasonable and natural routes. We show by both theoretical analysis and simulations that our medial axis based geometric routing scheme is scalable, produces short routes, achieves excellent load balancing, and is very robust to variations in the network model

Quantifiable isovist and graph-based measures for automatic evaluation of different area types in virtual terrain generation

© 2013 IEEE. This article describes a set of proposed measures for characterizing areas within a virtual terrain in terms of their attributes and their relationships with other areas for incorporating game designers\u27 intent in gameplay requirement-based terrain generation. Examples of such gameplay elements include vantage point, strongholds, chokepoints and hidden areas. Our measures are constructed on characteristics of an isovist, that is, the volume of visible space at a local area and the connectivity of areas within the terrain. The calculation of these measures is detailed, in particular we introduce two new ways to accurately and efficiently calculate the 3D isovist volume. Unlike previous research that has mainly focused on aesthetic-based terrain generation, the proposed measures address a gap in gameplay requirement-based terrain generation-the need for a flexible mechanism to automatically parameterise specified areas and their associated relationships, capturing semantic knowledge relating to high level user intent associated with specific gameplay elements within the virtual terrain. We demonstrate applications of using the measures in an evolutionary process to automatically generate terrains that include specific gameplay elements as defined by a game designer. This is significant as this shows that the measures can characterize different gameplay elements and allow gameplay elements consistent with the designers\u27 intents to be generated and positioned in a virtual terrain without the need to specify low-level details at a model or logic level, hence leading to higher productivity and lower cost

Quantifiable isovist and graph-based measures for automatic evaluation of different area types in virtual terrain generation

© 2013 IEEE. This article describes a set of proposed measures for characterizing areas within a virtual terrain in terms of their attributes and their relationships with other areas for incorporating game designers\u27 intent in gameplay requirement-based terrain generation. Examples of such gameplay elements include vantage point, strongholds, chokepoints and hidden areas. Our measures are constructed on characteristics of an isovist, that is, the volume of visible space at a local area and the connectivity of areas within the terrain. The calculation of these measures is detailed, in particular we introduce two new ways to accurately and efficiently calculate the 3D isovist volume. Unlike previous research that has mainly focused on aesthetic-based terrain generation, the proposed measures address a gap in gameplay requirement-based terrain generation-the need for a flexible mechanism to automatically parameterise specified areas and their associated relationships, capturing semantic knowledge relating to high level user intent associated with specific gameplay elements within the virtual terrain. We demonstrate applications of using the measures in an evolutionary process to automatically generate terrains that include specific gameplay elements as defined by a game designer. This is significant as this shows that the measures can characterize different gameplay elements and allow gameplay elements consistent with the designers\u27 intents to be generated and positioned in a virtual terrain without the need to specify low-level details at a model or logic level, hence leading to higher productivity and lower cost

Computing a discrete Morse gradient from a watershed decomposition

We consider the problem of segmenting triangle meshes endowed with a discrete scalar function f based on the critical points of f . The watershed transform induces a decomposition of the domain of function f into regions of influence of its minima, called catchment basins. The discrete Morse gradient induced by f allows recovering not only catchment basins but also a complete topological characterization of the function and of the shape on which it is defined through a Morse decomposition. Unfortunately, discrete Morse theory and related algorithms assume that the input scalar function has no flat areas, whereas such areas are common in real data and are easily handled by watershed algorithms. We propose here a new approach for building a discrete Morse gradient on a triangulated 3D shape endowed by a scalar function starting from the decomposition of the shape induced by the watershed transform. This allows for treating flat areas without adding noise to the data. Experimental results show that our approach has significant advantages over existing ones, which eliminate noise through perturbation: it is faster and always precise in extracting the correct number of critical elements