By L. Huang
This up to date moment version broadens the reason of rotational kinematics and dynamics — an important element of inflexible physique movement in three-d area and a subject matter of a lot better complexity than linear movement. It expands therapy of vector and matrix, and contains quaternion operations to explain and learn inflexible physique movement that are present in robotic keep watch over, trajectory making plans, 3D imaginative and prescient procedure calibration, and hand-eye coordination of robots in meeting paintings, and so forth. It positive factors up-to-date remedies of strategies in all chapters and case studies.
The textbook keeps its comprehensiveness in assurance and compactness in measurement, which make it simply available to the readers from multidisciplinary parts who are looking to clutch the major recommendations of inflexible physique mechanics that are often scattered in a number of volumes of conventional textbooks. Theoretical options are defined via examples taken from throughout engineering disciplines and hyperlinks to functions and extra complex classes (e.g. business robotics) are provided.
Ideal for college kids and practitioners, this ebook presents readers with a transparent route to figuring out inflexible physique mechanics and its importance in different sub-fields of mechanical engineering and comparable areas.
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Additional info for A Concise Introduction to Mechanics of Rigid Bodies: Multidisciplinary Engineering
Q/ D kqk D q p p p qq D qq D q20 C k! q/: • Inverse: q 1 D! q =kqk2 ; ! p0 ! p /. p0 ! q//=2: 24 1 Preliminaries on Vectors, Matrices, Complex Numbers and Quaternions • Unit quaternion and differential calculus: When kqk D 1, q is called a unit quaternion, which can be represented by q D cos Â C uO sin Â D euO Â ; q =k! q k. The quantity uO can be treated as a unit where cos Â D q0 and uO D ! vector or pure quaternion. t/O In the following example, various quaternion operations are demonstrated.
Q0 p0 ! p ; q0! p C p0! q C! q q T! 28) Note quaternion multiplication uses both the vector inner (dot) and outer (cross) products. If p and q are pure quaternions, then qp D ! q ! p; q! q D ! q T! qq D q2 D ! q D k! 4 Quaternions 23 In matrix form, Ä Ä ! 0 p D q: ! 3 3 q q0 I C Œ q p p0 I Œ! p Obviously, pq ¤ qp, unless Œ! q ! p D 0 or ! q ! p D 0. This means that ! q and ! p are parallel to each other. q0 ; q/; qp D qp; qq D qq D q20 C ! q: q T! q/ D kqk D q p p p qq D qq D q20 C k! q/: • Inverse: q 1 D!
21). When sÂ D 0, then Â D 0 or Â D ˙ . Â/ D I 3 ) and rO is undefined. Rr C I 3 /=2. Note that R r . Â/. This is an ambiguity in the equivalent/effective axis–angle representation of rotation. Â/ corresponding to the equivalent/effective representation of rotation [Eq. 12)]. Applying Taylor expansions of the sine and cosine functions, sin Â D 1 X . 2k C 1/Š 1 X x2k ; . 2k/Š kD0 and considering that ŒOr 2k D . 1/kC1 ŒOr 2 ; ŒOr 2kC1 D . Â/ D 1 X ŒOr k D eŒOr kŠ kD0 Â : These are the exponential coordinates for the rotation specified by an axis (Or) and an angle (Â).
A Concise Introduction to Mechanics of Rigid Bodies: Multidisciplinary Engineering by L. Huang