Convex Hulls: Complexity and Applications (a Survey)

Computational geometry is, in brief, the study of algorithms for geometric problems. Classical study of geometry and geometric objects, however, is not well-suited to efficient algorithms techniques. Thus, for the given geometric problems, it becomes necessary to identify properties and concepts that lend themselves to efficient computation. The primary focus of this paper will be on one such geometric problems, the Convex Hull problem

Shadow Computations using Robust Epsilon Visibility

Analytic visibility algorithms, for example methods which compute a subdivided mesh to represent shadows, are notoriously unrobust and hard to use in practice. We present a new method based on a generalized definition of extremal stabbing lines, which are the extremities of shadow boundaries. We treat scenes containing multiple edges or vertices in degenerate configurations, (e.g., collinear or coplanar). We introduce a robust epsilon method to determine whether each generalized extremal stabbing line is blocked, or is touched by these scene elements, and thus added to the line's generators. We develop robust blocker predicates for polygons which are smaller than epsilon. For larger values, small shadow features merge and eventually disappear. We can thus robustly connect generalized extremal stabbing lines in degenerate scenes to form shadow boundaries. We show that our approach is consistent, and that shadow boundary connectivity is preserved when features merge. We have implemented our algorithm, and show that we can robustly compute analytic shadow boundaries to the precision of our chosen epsilon threshold for non-trivial models, containing numerous degeneracies

AMP-CAD: Automatic Assembly Motion Planning Using C AD Models of Parts

Assembly with robots involves two kinds of motions, those that are point-to-point and those that are force/torque guided, the former kind of motions being faster and more amenable to automatic planning and the latter kind being necessary for dealing with tight clearances. In this paper, we describe an assembly motion planning system that uses descriptions of assemblies and CAD models of parts to automatically figure out which motions should be point-to-point and which motions should be force/torque guided. Our planner uses graph search over a potential field representation of parts to calculate candidate assembly paths. Given the tolerances of the parts and other uncertainties, these paths are then analyzed for the likelihood of collisions. Those path segments that are prone to collisions are then marked for execution under force/torque control. The calculation of the various motions is facilitated by an object-oriented and feature-based assembly representation. A highlight of this representation is the manner in which tolerance information is taken into account: Representation of, say, a part contains a pointer to the boundary representation of the part in its most material condition form. As first defined by Requicha, the most material condition form of a geometric entity is obtained by expanding all the convexities and shrinking all the concavities by relevant tolerances. An integral part of the assembly motion planner is the execution unit. Residing in this unit is knowledge of the different types of automatic EDR (error detection and recovery) strategies. Therefore, during the execution of the force/torque guided motion, this unit invokes the EDR strategies appropriate to the geometric constraints relevant to the motion. This system, called AMP-CAD, has been experimentally verified using a Cincinnati Milacron T3-726 robot and a Puma 762 robot on a variety of assemblies

Functionality and performance: two important considerations when implementing topology in 3D.

This thesis contributes to the understanding of the use of topology in analysing 3D spatial data, focussing in particular on two aspects of the problem - what binary topological analysis functionality is required in a commercial 3D Geographical Information System (GIS), and how should this functionality be implemented to achieve the most efficient query performance. Topology is defined as the identification of spatial relationships between adjacent or neighbouring objects. The first stage of this research, a review of applications of topology, results in a generic list of requirements for topology in 3D. This was carried out in parallel with a review of topological frameworks and the relationships identified by one of the frameworks, Egenhofer and Herring's 9-Intersection, selected for implementation. Three generic binary relationship queries are identified (Find Objects with a Specific Relationship, Find Intersecting Objects and What Relationship is there Between These Objects) and a mechanism described to allow these to be adapted to specific application terminology. Approaches to the implementation of 3D binary topological queries include the use of data structures and an As-Required calculation, where computational geometry algorithms are run to determine relationships each time the user runs a query. The Three-Dimensional Formal Data Structure (3DFDS) was selected as a representative example of a Boundary-Representation (B- Rep) structure in GIS. Given the number of joins to be traversed when identifying binary relationships from a B-Rep structure, along with the requirement to query additional containment exception tables, an alternate structure, the Simplified Topological Structure (STS), was proposed to improve binary query performance. Binary relationship queries were developed and comparative performance tests carried out against 3DFDS, STS and a Proxy for the As-Required calculation, using a 1.08 million object test dataset. Results show that STS provides a significant performance improvement over 3DFDS. No definitive conclusion could be drawn when comparing STS with the Proxy for the As-Required approach

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Contact and HiL interaction in multibody based machinery simulators

[Abstract] Multibody simulators allow to predict and evaluate the motion of machines and mechanisms under the action of the user and the interaction with the simulated environment. Interactive simulators guided by a human or a piece of hardware must be efficient enough to compute the state of the system in real time. ?erefore, employing fast and sufficiently accurate techniques is a must. In this work, generic tools for the implementation of this kind of simulators are provided. Efficient multibody formulations are reviewed for implementing real-time simulators. ?e index-3 Augmented Lagrange formulation with projections of velocities and accelerations is selected, due to its efficiency and stability. ?e integration of the equations of motion follows the Generalized-a method, which provides high-frequency dissipation, and can be unconditionally stable and secondorder accurate if suitable integrator parameters are chosen. Contact modeling and detection is essential for computing the interaction among the mechanisms and the simulated environment. Normal and tangential contact force models are presented. For the normal contact, a Hertz-type Hunt- Crossley model is chosen. ?e tangential force model is based on Coulomb’s law, and includes stiction and viscous friction effects. Both models were compared with the output of the Bowden-Leben stick-slip experiment. A real-time, simplified terrain model featuring digging forces for excavator simulators is also discussed. Several techniques are shown for detecting colliding bodies at run-time. ?e collision detection process is divided into two stages. ?e first one is a broad range and coarse grained process, where potentially colliding pairs of objects are discovered. Spatial and hierarchical division techniques as Octrees, BSP-trees and Directed Acyclic Graphs are presented for this purpose. In the second stage, fine-detailed contact properties are computed from each pair of bodies. Several models are presented for testing object enclosing volumes or more complex surfaces discretized as triangular meshes. State-of-the-art, Commercial Off ?e Shelf hardware devices are presented as the physical foundation of a simulator. Industrial-quality controllers, projection screens and audio devices are reviewed for this purpose. ?e implementation details for the use of those devices are also considered. Network communication procedures between the simulator and monitoring nodes are discussed, too. Finally, a particular implementation of all the techniques described in previous chapters is presented in the form of an interactive excavator simulator, which features all the degrees of freedom of the machine, and is able to perform earthmoving operations in a realistic environment. Monitoring capabilities are also available, and any training session can be defined by user scripts. ?e techniques described in this document constitute a generic and efficient compendium of algorithms that are well-fi?ed for medium or low-end computational systems, as desktop or even laptop computers

Rock joint systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1985.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.Bibliography: p. 745-764.by William Simon Dershowitz.Ph.D

A stochastic approach to path planning in the Weighted-Region Problem

Planning efficient long-range movement is a fundamental requirement of most military operations. Intelligent mobile autonomous vehicles designed for battlefield support missions must have this capability. We propose an efficient heuristic algorithm for planning near-optimal high-level routes through complex terrain maps, modeled by the Weighted-Region Problem (WRP). The algorithm is driven by our adaptation of the combinatorial optimization technique called simulated annealing. The WRP provides a cartographically powerful representation for planar maps. Terrain features are modeled by polygonal homogenous-cost regions. A cost efficient assigned to each region indicates the relative cost per unit distance for movement in the region by a point agent no the ground. Region cost coefficients are assumed to be invariant with direction of movement. Given a start and a goal point, a solution (not necessarily optimal) is a set of piecewise-linear connected segments, spanning from start to goal. The cost of a solution path is the sum of the weighted lengths of all segments in the path, where the weighted length of each segment is the product of it Euclidian length and the cost coefficient of the region it crosses. Ideally, we seek the least cost path. However, as problem instances approach the actual complexity of the battlefield, faster solutions become more desirable than absolute optimality. We introduce heuristics designed to replace the search space independently of start and goal locations, thus allowing map preprocessing. We use other heuristics to improve the efficiency of local cost function optimization as well as the annealing process itself.http://archive.org/details/stochasticapproa00kindMajor, United States ArmyApproved for public release; distribution is unlimited