PPT-Quaternion

靜宜大學資工系 蔡奇偉副教授 2010 大綱 History of Quaternions Definition of Quaternion Operations Unit Quaternion Operation Rules Quaternion Transforms Matrix Conversion History of . LaV iola Jr Bro wn Uni ersity echnology Center for Adv anced Scienti57346c Computing and isualization PO Box 1910 Pro vidence RI 02912 USA Emailjjlcsbrownedu Abstract The unscented Kalman 57346lter is superior alter na ti to the extended Kalman 5734 with Quaternion Curves Shoemaker Singer Company Link Flight Simulation Division CT bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best Open-source C++ Library Implementation. Notation and Identities. Details:. . Rotation matrices, special orthogonal group, quaternions, calculation of quaternion vector rotation, parallel, antiparallel, robust rotation matrix to quaternion conversions, and vice versa.. Euler Theorem + Quaternions . Representing a Point 3D. A three-dimensional point. . A. is a reference coordinate system here. Rotation along the . Z axis. In general:. Using Rotation Matrices. 750. Texture. , Microstructure & Anisotropy. A.D. (Tony) Rollett,. . S. R. Wilson. Rodrigues. . vectors. ,. unit . Quaternions. Last revised: . 2. nd. Jan. 2015. Briefly describe rotations/orientations. Review of some concepts:. trigonometry. aliasing. coordinate systems. homogeneous coordinates. matrices, . quaternions. Vectors. v = . ai. + . bj. + ck. Describes point or displacement in n-dimensional space. The Phoenix Bird of Mathematics. Herb Klitzner. June 1, 2015. Presentation to:. New York Academy of Sciences, Lyceum Society. © 2015, Herb Klitzner. http://quaternions.klitzner.org. . The Phoenix Bird. The Phoenix Bird of Mathematics. Herb Klitzner. June 1, . 2015. Presentation to:. New York Academy of Sciences, Lyceum . Society. © 2015, Herb Klitzner. The Phoenix Bird. CONTENTS. INTRODUCTION . APPLICATIONS. QUATERNIONS. Spring 2015. Dr. Michael J. Reale. INTRODUCTION. Quaternions invented by Sir William Rowan Hamilton in 1843. Developed as extension to complex numbers. Introduced into computer graphics by Ken Shoemake in 1985. Euler Theorem + Quaternions . Representing a Point 3D. A three-dimensional point. . A. is a reference coordinate system here. Rotation along the . Z axis. In general:. Using Rotation Matrices. provides an approach to computing the average quaternion by min- imizing a quaternion cost function that is equivalent equivalent Motivated by [l] and extending its results, this Note derives an algor Lecture 4, Part B:. Orientation Representation. 1. Rotation Representation. 2. We saw that for computer graphics/games, the three most commonly used transforms are: . translation. , . rotation. . and . This book discusses all spacecraft attitude control-related topics: spacecraft (including attitude measurements, actuator, and disturbance torques), modeling, spacecraft attitude determination and estimation, and spacecraft attitude controls. Unlike other books addressing these topics, this book focuses on quaternion-based methods because of its many merits. The book lays a brief, but necessary background on rotation sequence representations and frequently used reference frames that form the foundation of spacecraft attitude description. It then discusses the fundamentals of attitude determination using vector measurements, various efficient (including very recently developed) attitude determination algorithms, and the instruments and methods of popular vector measurements. With available attitude measurements, attitude control designs for inertial point and nadir pointing are presented in terms of required torques which are independent of actuators in use. Given the required control torques, some actuators are not able to generate the accurate control torques, therefore, spacecraft attitude control design methods with achievable torques for these actuators (for example, magnetic torque bars and control moment gyros) are provided. Some rigorous controllability results are provided.The book also includes attitude control in some special maneuvers, such as orbital-raising, docking and rendezvous, that are normally not discussed in similar books. Almost all design methods are based on state-spaced modern control approaches, such as linear quadratic optimal control, robust pole assignment control, model predictive control, and gain scheduling control. Applications of these methods to spacecraft attitude control problems are provided. Appendices are provided for readers who are not familiar with these topics. Notation and Identities. Details:. . Rotation matrices, special orthogonal group, quaternions, calculation of quaternion vector rotation, parallel, antiparallel, robust rotation matrix to quaternion conversions, and vice versa..