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

Roots of bivariate polynomial systems via determinantal representations

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coefficients, and consists of matrices with asymptotic order $n^2/4$, where $n$ is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order $n^2/6$. The resulting method to compute the roots of a system of two bivariate polynomials is competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.Comment: 22 pages, 9 figure

Predictive and Multi-rate Sensor-Based Planning under Uncertainty

Email Print Request Permissions In this paper, a general formulation of a predictive and multirate (MR) reactive planning method for intelligent vehicles (IVs) is introduced. The method handles path planning and trajectory planning for IVs in dynamic environments with uncertainty, in which the kinodynamic vehicle constraints are also taken into account. It is based on the potential field projection method (PFP), which combines the classical potential field (PF) method with the MR Kalman filter estimation. PFP takes into account the future object trajectories and their associated uncertainties, which makes it different from other look-ahead approaches. Here, a new PF is included in the Lagrange-Euler formulation in a natural way, accounting for the vehicle dynamics. The resulting accelerations are translated into control inputs that are considered in the estimation process. This leads to the generation of a local trajectory in real time (RT) that fully meets the constraints imposed by the kinematic and dynamic models of the IV. The properties of the method are demonstrated by simulation with MATLAB and C++ applications. Very good performance and execution times are achieved, even in challenging situations. In a scenario with 100 obstacles, a local trajectory is obtained in less than 1 s, which is suitable for RT applications

Adaptive color spaces based on multivariate Gaussian distributions for color image segmentation

We formulate an adaptive color space for segmenting all image into the two classes "object of interest" and "background" by using well-established methods from statistical pattern recognition. Both classes are modeled by a multivariate Gaussian distribution whose actual parameters are estimated via the Expectation Maximization (EM) algorithm. The output grayscale feature image is derived as the distance of each pixel's color to the decision boundary which is shaped bewteen the two class models. Based on this feature image, which provides a maximum discriminatory power with respect to the underlying model assumptions, the actual segmentation can be performed with appropriate methods from grayscale image processing. This adaptive color space is a practical tool for homogeneously colored scenes, as they appear, e.g., in microscopic images of biotechnical fundamental research

Continuous collision detection for ellipsoids

We present an accurate and efficient algorithm for continuous collision detection between two moving ellipsoids. We start with a highly optimized implementation of interference testing between two stationary ellipsoids based on an algebraic condition described in terms of the signs of roots of the characteristic equation of two ellipsoids. Then we derive a time-dependent characteristic equation for two moving ellipsoids, which enables us to develop a real-time algorithm for computing the time intervals in which two moving ellipsoids collide. The effectiveness of our approach is demonstrated with several practical examples. © 2006 IEEE.published_or_final_versio

Between Algorithm and Model: Different Molecular Surface Definitions for the Poisson-Boltzmann based Electrostatic Characterization of Biomolecules in Solution

The definition of a molecular surface which is physically sound and computationally efficient is a very interesting and long standing problem in the implicit solvent continuum modeling of biomolecular systems as well as in the molecular graphics field. In this work, two molecular surfaces are evaluated with respect to their suitability for electrostatic computation as alternatives to the widely used Connolly-Richards surface: the blobby surface, an implicit Gaussian atom centered surface, and the skin surface. As figures of merit, we considered surface differentiability and surface area continuity with respect to atom positions, and the agreement with explicit solvent simulations. Geometric analysis seems to privilege the skin to the blobby surface, and points to an unexpected relationship between the non connectedness of the surface, caused by interstices in the solute volume, and the surface area dependence on atomic centers. In order to assess the ability to reproduce explicit solvent results, specific software tools have been developed to enable the use of the skin surface in Poisson-Boltzmann calculations with the DelPhi solver. Results indicate that the skin and Connolly surfaces have a comparable performance from this last point of view

Personalised body counter calibration using anthropometric parameters

This book describes the development of a new method for personalisation of efficiency factors in partial body counting. Its achieved goal is the quantification of uncertainties in those factors due to variation in anatomy of the measured persons, and their reduction by correlation with anthropometric parameters. The method was applied to a detector system at the In Vivo Measurement Laboratory at Karlsruhe Institute of Technology using Monte Carlo simulation and computational phantoms

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