sv01

Two libraries, the same two quaternions, two different rotations, and neither library is wrong. One identity explains all of it. The JPL quaternion product is exactly the Hamilton product with its arguments reversed, verified here to three and a half parts in ten to the seventeenth, and that single fact accounts for the entire class of bug. The video separates the three choices that are constantly mistaken for one: multiplication order, scalar first versus scalar last storage, and whether a quaternion rotates the vector or the frame. Each is independent, each can be right while another is wrong, and the damage is measured rather than asserted: a storage swap moves a random unit quaternion by a hundred and one degrees, and only about one mixed pair in thirteen hundred lands within five degrees of correct. It ends with a four step test that identifies an unknown library's conventions in about five minutes with no access to its source. This is one of a series of standalone videos, each answering one question completely in about ten minutes, for engineers who work with rotations and have been burned by them. Linear algebra and ordinary differential equations are enough; no course on smooth manifolds is assumed. Every number stated in this video was recomputed and checked before it was printed, including deliberate control tests built to fail, and every convention is stated on the slide wherever the literature disagrees. If you want the full treatment, these videos sit alongside Applied Lie Groups for Engineers, a twenty lecture course on the same channel. Reading: Sola, Quaternion kinematics for the error-state Kalman filter, sections 1 to 3; Sola, Deray and Atchuthan, micro Lie theory, section II; Shuster, A survey of attitude representations, 1993. Texts, all free online: Barfoot, State Estimation for Robotics, 2nd edition; Sola, Deray and Atchuthan, A micro Lie theory for state estimation in robotics (arXiv 1812.01537); Sola, Quaternion kinematics for the error-state Kalman filter; Lynch and Park, Modern Robotics. Chapters 0:00 Introduction 0:29 The first way to be wrong: multiplication order 1:35 Hamilton's product, and the one the JPL literature uses 2:48 The identity that explains the entire video 3:54 Why the wrong answer is never a rounding error 4:56 Three more ways, and how to catch all four 6:15 A third choice: the vector moves, or the frame does 7:31 Not a choice at all: a quaternion and its negative 8:46 A five minute test for an unknown library 10:02 Four choices, one library, and how to stop guessing