Some of the confusions concerning quaternions as they are employed in spacecraft attitude work are discussed. The order of quaternion multiplication is discussed in terms of its historical development and its consequences for the quaternion imaginaries. The different formulations for the quaternions are also contrasted. It is shown that the three Hamilton imaginaries cannot be interpreted as the basis of the vector space of physical vectors but only as constant numerical column vectors, the autorepresentation of a physical basis.
Quaternions Rotation Sequences Kuipers Pdf Download
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The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.
Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the nineteenth century--a feat he celebrated by carving the founding equations into a stone bridge--mathematicians and engineers have been fascinated by these mathematical objects. Today, they are used in applications as various as describing the geometry of spacetime, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J. B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations. The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.
Just as complex numbers handle rotations in the plane, so quaternions are meant to handle rotations in three-dimensional space. But, misled by the analogy with the two-dimensional case, where multiplication by a single unit complex number produces a rotation, Hamilton assumed such rotations could be represented by a single multiplication qv, where v is any vector and q is a unit quaternion. This one-sided multiplication works only for a special case: rotations orthogonal to the
vector, about its axis. Thus from the very evening of 16 October 1843 Hamilton bungled the quaternion account of rotation, which should be dealt with by pre- and postmultiplication by a quaternion rvr1, with r defined not (as in Hamilton) by the angle of the rotation but by its half-angle. The geometry of composing rotations and the need to deal in half-angles had already been properly done in 1840 by a French mathematician who was also a socialist banker, Benjamin Olinde Rodrigues, but had (not unsurprisingly) escaped Hamilton's notice. Particularly insightful and detailed accounts of Hamilton's mistake and Rodrigues's success are given by Altmann (1986,1989,1992), who deserves much credit for showing that the neglect of quaternions was for a long time excessive, as well as for investigating the hitherto highly obscure Rodrigues (Altmann and Ortiz, 2005).1 2ff7e9595c
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