An algebraic condition and an algorithm for the internal contact between two ellipsoids

4th International Conference on Discrete Element Methods, Brisbane, AUSTRALIA, AUG, 2007International audiencePurpose - The purpose of this paper is to present a new method for the detection and resolution of the contact point between two ellipsoids. Numerical simulations of ellipsoidal particles in a rotary cylinder are also presented. Desigri/methodology/approach - An algebraic condition is developed for the internal contact between two ellipsoids and an efficient contact detection algorithm for overlapping ellipsoids is implemented. Findings - This method was found to have the advantages of effectiveness and speed in the detection and resolution of the contact point. Originality/value - The dynamics of granular materials are of great importance in many industries dealing with powders and grains, such as pharmaceutical, chemical, and food industries. The main difficulty of such simulations is the excessive CPU time required for a large number of particles. In the discrete element method, contact detection between grains is the most expensive step in solving a nonlinear system for determination of the contact point, the normal vector and the overlap distance between ellipsoids. The numerical behavior and the optimization of the new algorithm presented in this paper are important also

Intersection Testing between an Ellipsoid and an Algebraic Surface

International audienceThis paper presents a new method on the intersection testing problem between an ellipsoid and an algebraic surface. In the new method, the testing problem is turned into a new testing problem whether a univariate polynomial has a positive or negative real root. Examples are shown to illustrate the robustness and efficiency of the new method

Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification

We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives

On the geometry of geared 5-bar motion

A new approach is adopted to study the geometry of the coupler curves associated to geared 5-bar motion. The key idea is to think of a configuration of the mechanism as a point in a higher -dimensional configuration space; the family of all configurations is then represented by an algebraic curve in that space. Coupler curves appear naturally as projections of this curve, so their properties can be deduced by projection, independent of any explicit knowledge of their equations. Introduction The present paper has its genesis in the general problem of elucidating the algebraic geometry of coupler curves for planar mechanisms. Much depends on knowing the number and nature of the singular points, so one seeks a better understanding of how these arise. A first small step in this direction was taken in Given the above context, the discussion in [4] of geared 5-bar motion assumes particular interest, in view of the authors' comment that here again the coupler curve has just five singular points, lying on a conic. In this paper we take the same viewpoint of geared 5-bar motion as was taken in [6] for the planar 4-bar. The net result is that one is able to say rather more about the geometry of coupler curves than appeared in (/) In [6] it was shown that the Grashof equations correspond precisely to the natural geometric condition that the linkage curve has a singularity off the hyperplane at infinity. The latter condition makes perfect sense for any planar mechanism so provides a sensible general definition of the term Grashof equation. In particular, we can adopt this viewpoint for th

Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

International audienceArrangements of geometric objects refer to the spatial partitions formed by the objects and they serve as an underlining structure of motion design, analysis and planning in CAD/CAM, robotics, molecular modeling, manufacturing and computer-assisted radio-surgery. Arrangements are especially useful to collision detection, which is a key task in various applications such as computer animation , virtual reality, computer games, robotics, CAD/CAM and computational physics. Ellipsoids are commonly used as bounding volumes in approximating complex geometric objects in collision detection. In this paper we present an in-depth study on the arrangements formed by two ellipsoids. Specifically, we present a classification of these arrangements and propose an efficient algorithm for determining the arrangement formed by any particular pair of ellipsoids. A stratification diagram is also established to show the connections among all the arrangements formed by two ellipsoids. Our results for the first time elucidate all possible relative positions between two arbitrary ellipsoids and provides an efficient and robust algorithm for determining the relative position of any two given ellipsoids, therefore providing the necessary foundation for developing practical and trustworthy methods for processing ellipsoids for collision analysis or simulation in various applications

Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the Betti numbers as well as on the number of different stable homotopy types of certain fibers of semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations. The first algorithm computes the number of connected components and the first Betti number of a semi-algebraic set defined by compact objects in $\mathbb{R}^k$ which are simply connected. This algorithm improves the well-know method using a triangulation of the semi-algebraic set. Moreover, the algorithm has been efficiently implemented which was not possible before. The second algorithm computes efficiently the real intersection of three quadratic surfaces in $\mathbb{R}^3$ using a semi-numerical approach.Comment: PhD thesis, final version, 109 pages, 9 figure

타원 로봇의 충돌 회피를 위한 속도 기반의 지역 경로 계획 방법

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 이범희.Collision-free motion planning has been hierarchically decomposed into two parts: global and local planners. While the former generates the shortest path to the goal from global environmental information, the latter modifies the path from the global one by considering unexpected dynamic obstacles and motion constraints of mobile robots. In the local navigation problem, robots and obstacles have been approximated by simple geometric objects in order to decrease the computation time. They have been generally enclosed by circles due to its simplicity in collision detection. However, this approximation becomes overly conservative if the objects are elongated, which leads the robots to travel longer paths than necessary to avoid collisions. This dissertation presents a velocity-based approach to address the local navigation problem of anisotropic mobile robots bounded by ellipses. Compared with the other geometries, Löwner ellipse, the minimum area bounding ellipse, provides more compact representation for robots and obstacles in a 2D plane, but the collision detection between them is more complicated. Hence, it is first investigated under what conditions a collision between two ellipses occurs. To this end, the configuration space framework and an algebraic approach are introduced. In the former method, it is found that an elliptic robot can be regarded as a circular robot with radius equal to its minor radius by adequately controlling its orientation. In the latter method, the interior-disjoint condition between two ellipses is characterized by four inequalities. Next, a velocity-based approach is suggested on the basis of the collision detection so that an elliptic robot moves to its goal without collisions with obstacles. The proposed algorithm is decomposed into two phases: linear and angular motion planning. In the first phase, the ellipse-based velocity obstacle (EBVO) is defined as the set of linear velocities of a robot that would cause a collision within a finite time horizon. Furthermore, strategies for determining a new linear velocity with the EBVO are explained. In the second phase, the angular velocity is selected with which the robot can circumvent the obstacle blocking the path to the goal with the minimum deviation. Finally, the obstacle avoidance method was extended for multi-robot collision avoidance on the basis on the concept of reciprocity. The concept of hybrid reciprocal velocity obstacles is adopted in the part of linear motion planning, and the collision-free reciprocal rotation angles are calculated in the part of angular motion planning on the assumption that if one robot rotates, then the other robot may rotate equally or equally opposite. The proposed algorithm was validated in simulations for various scenarios in terms of travel time and distance. It was shown that it outperformed the methods that enclosed robots and obstacles by circles, by ellipses without rotation, and by polygons with rotation. In addition, it was shown that the computation time of the proposed method was much smaller than the sampling time, which means that it is fast enough for real-time applications.Chapter 1 Introduction 1 1.1 Background of the Problem 1 1.2 Statement of the Problem 5 1.3 Contributions 10 1.4 Organization 11 Chapter 2 Literature Review 13 2.1 Bounding Ellipsoid 13 2.2 Collision Detection between Ellipsoids 15 2.3 Velocity-based Local Navigation 18 Chapter 3 Collision Detection 23 3.1 Introduction 23 3.2 Problem Formulation 25 3.3 Configuration Space Obstacle 25 3.4 Algebraic Condition for the Interior-disjoint of Two Ellipses 34 3.5 Summary 50 Chapter 4 Obstacle Avoidance 51 4.1 Introduction 51 4.2 Problem Formulation and Approach 53 4.3 Preliminaries: Properties of C-obstacles for an Elliptic Robot 56 4.3.1 Tangent lines to C-obstacle 56 4.3.2 Closest point on the outline of C-obstacle 63 4.4 Ellipse-based Velocity Obstacles 65 4.5 Selection of Collision-free Linear Velocity 71 4.5.1 Conservative Approximation of the EBVOs 72 4.5.2 New Linear Velocity Selection with Multiple Obstacles 77 4.6 Collision-free Rotation Angles 81 4.6.1 The Shortest Time-to-contact 81 4.6.2 Collision-free Interval of the Rotation Angles 82 4.7 Selection of Collision-free Angular Velocity 89 4.7.1 Preferred Angular Velocities 89 4.7.2 New Angular Velocity Selection 91 4.8 Summary 93 Chapter 5 Multi-Robot Collision Avoidance 95 5.1 Introduction 95 5.2 Problem Formulation 97 5.3 Ellipse-based Reciprocal Velocity Obstacles 98 5.4 Collision-free Reciprocal Rotation Angles 103 5.4.1 Candidates of the First Contact Rotation Angle 108 5.4.2 Updating the Candidates Sets 116 5.4.3 Calculation of Collision-free Reciprocal Rotation Angles 117 5.4.4 An Example 118 5.5 Summary 123 Chapter 6 Implementation and Simulations 125 6.1 Implementation Setups 125 6.2 Obstacle Avoidance 126 6.2.1 Line scenario of a robot and an obstacle 127 6.2.2 Multiple moving obstacles scenario 135 6.2.3 Pedestrians avoidance scenario 144 6.3 Multi-Robot Collision Avoidance 148 6.3.1 Chicken scenario 149 6.3.2 Circle scenario 155 Chapter 7 Conclusion 165 Bibliography 171 초록 191Docto