BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning

The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Comparison of Current Gravity Estimation and Determination Models

This paper will discuss the history of gravity estimation and determination models while analyzing methods that are in development. Some fundamental methods for calculating the gravity field include spherical harmonics solutions, local weighted interpolation, and global point mascon modeling (PMC). Recently, high accuracy measurements have become more accessible, and the requirements for high order geopotential modeling have become more stringent. Interest in irregular bodies, accurate models of the hydrological system, and on-board processing has demanded a comprehensive model that can quickly and accurately compute the geopotential with low memory costs. This trade study of current geopotential modeling techniques will reveal that each modeling technique has a unique use case. It is notable that the spherical harmonics model is relatively accurate but poses a cumbersome inversion problem. PMC and interpolation models, on the other hand, are computationally efficient, but require more research to become robust models with high levels of accuracy. Considerations of the trade study will suggest further research for the point mascon model. The PMC model should be improved through mascon refinement, direct solutions that stem from geodetic measurements, and further validation of the gravity gradient. Finally, the potential for each model to be implemented with parallel computation will be shown to lead to large improvements in computing time while reducing the memory cost for each technique.Aerospace Engineering and Engineering Mechanic

High-Fidelity Gravity Modeling Applied to Spacecraft Trajectories and Lunar Interior Analysis

As the complexity and boldness of emerging mission proposals increase, and with the rapid evolution of the available computational capabilities, high-accuracy and high-resolution gravity models and the tools to exploit such models are increasingly attractive within the context of spaceflight mechanics, mission design and analysis, and planetary science in general. First, in trajectory design applications, a gravity representation for the bodies of interest is, in general, assumed and exploited to determine the motion of a spacecraft in any given system. The focus is the exploration of trajectories in the vicinity of a system comprised of two small irregular bodies. Within this context, the primary bodies are initially modeled as massive ellipsoids and tools to construct third-body trajectories are developed. However, these dynamical models are idealized representations of the actual dynamical regime and do not account for any perturbing effects. Thus, a robust strategy to maintain a spacecraft near reference third-body trajectories is constructed. Further, it is important to assess the perturbing effect that dominates the dynamics of the spacecraft in such a region as a function of the baseline orbit. Alternatively, the motion of the spacecraft around a given body may be known to extreme precision enabling the derivation of a very high-accuracy gravity field for that body. Such knowledge can subsequently be exploited to gain insight into specific properties of the body. The success of the NASA\u27s GRAIL mission ensures that the highest resolution and most accurate gravity data for the Moon is now available. In the GRAIL investigation, the focus is on the specific task of detecting the presence and extent of subsurface features, such as empty lava tubes beneath the mare surface. In addition to their importance for understanding the emplacement of the mare flood basalts, open lava tubes are of interest as possible habitation sites safe from cosmic radiation and micrometeorite impacts. Tools are developed to best exploit the rich gravity data toward the numerical detection of such small features

PuzzleFlex: kinematic motion of chains with loose joints

This paper presents a method of computing free motions of a planar assembly of rigid bodies connected by loose joints. Joints are modeled using local distance constraints, which are then linearized with respect to configuration space velocities, yielding a linear programming formulation that allows analysis of systems with thousands of rigid bodies. Potential applications include analysis of collections of modular robots, structural stability perturbation analysis, tolerance analysis for mechanical systems, and formation control of mobile robots.Comment: Accepted at the 2020 IEEE International Conference on Robotics and Automation (ICRA

Grasping and Assembling with Modular Robots

A wide variety of problems, from manufacturing to disaster response and space exploration, can benefit from robotic systems that can firmly grasp objects or assemble various structures, particularly in difficult, dangerous environments. In this thesis, we study the two problems, robotic grasping and assembly, with a modular robotic approach that can facilitate the problems with versatility and robustness. First, this thesis develops a theoretical framework for grasping objects with customized effectors that have curved contact surfaces, with applications to modular robots. We present a collection of grasps and cages that can effectively restrain the mobility of a wide range of objects including polyhedra. Each of the grasps or cages is formed by at most three effectors. A stable grasp is obtained by simple motion planning and control. Based on the theory, we create a robotic system comprised of a modular manipulator equipped with customized end-effectors and a software suite for planning and control of the manipulator. Second, this thesis presents efficient assembly planning algorithms for constructing planar target structures collectively with a collection of homogeneous mobile modular robots. The algorithms are provably correct and address arbitrary target structures that may include internal holes. The resultant assembly plan supports parallel assembly and guarantees easy accessibility in the sense that a robot does not have to pass through a narrow gap while approaching its target position. Finally, we extend the algorithms to address various symmetric patterns formed by a collection of congruent rectangles on the plane. The basic ideas in this thesis have broad applications to manufacturing (restraint), humanitarian missions (forming airfields on the high seas), and service robotics (grasping and manipulation)

An Exact Representation of Polygonal Objects by C1-continuous Scalar Fields Based on Binary Space Partitioning

The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of function-based modelling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples