By Edward Y. L. Gu
This booklet offers readers with an excellent set of diverse and crucial instruments for the theoretical modeling and keep an eye on of advanced robot structures, in addition to for electronic human modeling and life like movement iteration. Following a finished advent to the basics of robot kinematics, dynamics and keep an eye on structures layout, the writer extends robot modeling strategies and movement algorithms to a miles higher-dimensional, better scale and extra refined study region, specifically electronic human modeling. many of the tools are illustrated via MATLAB codes and pattern graphical visualizations, providing a different closed loop among conceptual realizing and visualization.
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Extra info for A Journey from Robot to Digital Human: Mathematical Principles and Applications with MATLAB Programming
24 2 Mathematical Preliminaries With the above interesting properties owned by the unity skew-symmetric matrix K, we now develop a realistic rotation process, called the k–φ procedure. First, let φk ∈ R3 be deﬁned as a 3 by 1 vector. 6) yields exp(φK) = R ∈ SO(3). 7) The result of the exponential mapping is a rotation matrix that should be equivalent to the rotation by φ about the k-axis. 7) into Taylor series and noticing the above properties of K, we obtain φ2 2 φ3 3 φ4 4 K + K + K ··· 2! 3! 4!
20) In fact, this equation can be justiﬁed without diﬃculty through some major properties for a square matrix in linear algebra. First, a square matrix A can always be diagonalized or Jordan-decomposed via a similarity transformation P as follows: ⎞ ⎛ λ1 X .. ⎠, P −1 AP = ⎝ . O λn where λ1 through λn are the eigenvalues of A and the upper-right triangle block X may have some 1’s based on the Jordan decomposition, but the elements at the lower-left triangle corner are all zeros. 4 The Dual Number, Dual Vector and Their Algebras ⎛ 37 eλ1 ..
In contrast to rotation, a translation of either a vector tail-point or a coordinate frame is very straightforward without any complication, and we can simply use the addition of 3D vectors to perform any successive translations. ) motion in 3D space, one may desire to deﬁne a uniﬁed transformation to perform both translation and rotation simultaneously for a coordinate frame. Such uniﬁed 6-dimensional transformations will span the following Special Euclidean Space: SE(3) = (p, R) | p ∈ R3 , R ∈ SO(3) = R3 × SO(3), which is also a Lie group .
A Journey from Robot to Digital Human: Mathematical Principles and Applications with MATLAB Programming by Edward Y. L. Gu