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

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