By L. Huang
This up to date moment variation broadens the reason of rotational kinematics and dynamics — crucial element of inflexible physique movement in third-dimensional house and a subject matter of a lot higher complexity than linear movement. It expands therapy of vector and matrix, and comprises quaternion operations to explain and research inflexible physique movement that are present in robotic regulate, trajectory making plans, 3D imaginative and prescient method calibration, and hand-eye coordination of robots in meeting paintings, and so on. It beneficial properties up-to-date remedies of thoughts in all chapters and case studies.
The textbook keeps its comprehensiveness in insurance and compactness in dimension, which make it simply obtainable to the readers from multidisciplinary components who are looking to take hold of the major strategies of inflexible physique mechanics that are frequently scattered in a number of volumes of conventional textbooks. Theoretical thoughts are defined via examples taken from throughout engineering disciplines and hyperlinks to functions and extra complicated classes (e.g. commercial robotics) are provided.
Ideal for college kids and practitioners, this publication offers readers with a transparent route to realizing inflexible physique mechanics and its value in different sub-fields of mechanical engineering and similar areas.
Read or Download A Concise Introduction to Mechanics of Rigid Bodies: Multidisciplinary Engineering PDF
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Extra resources for A Concise Introduction to Mechanics of Rigid Bodies: Multidisciplinary Engineering
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! 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!
19). 19). q/ D ; ˛ D q0 C iq3 ; ˇ D q2 C iq1 ; D q2 C iq1 ; ı D q0 iq3 : ı It is a 2 2 matrix expression of a quaternion. It can be treated as an extension of a column vector, or one column matrix representing a complex number. Define 2 D4 1 2 3 5; 3 where below: 1, 2, and 3 are Pauli matrices as defined in Chap. 3 Orientation 47 Define the “inner product” between ! q and : ! q D 3 X qk k : kD1 Let q D erOÂ=2 . q/ D q0 I 2 C i 3 X qk k D q0 I 2 C i.! Or /Â=2 : Comparing it to q D erOÂ=2 , the similarity between their forms is obvious.
The rotational angles are ˛, ˇ, and , called the roll(R), pitch (P), and yaw (Y) angles. For simplicity and clarity of presentation, the immediate frames appearing in the rotations are not shown. Following the rules to derive the rotation matrix for the orientation of successive rotations relative to a fixed frame in Sect. ˇ/Rz . 29) 44 2 Orientation and Position Representation Fig. 11 RPY angles zˆA yˆA xˆA Note that this is the same as the rotation matrix of Z Y X angles, which means that the same orientation will be reached either by three rotations (X Y Z) about the axes of a fixed frame or by three rotations in opposite order (Z Y X) about the axes of the moving frames.