By Edward Y. L. Gu

ISBN-10: 3642390463

ISBN-13: 9783642390463

This booklet offers readers with an outstanding set of different and crucial instruments for the theoretical modeling and regulate of advanced robot structures, in addition to for electronic human modeling and practical movement iteration. Following a complete advent to the basics of robot kinematics, dynamics and keep watch over structures layout, the writer extends robot modeling methods and movement algorithms to a miles higher-dimensional, higher scale and extra subtle study zone, specifically electronic human modeling. lots of the tools are illustrated through MATLAB codes and pattern graphical visualizations, supplying a distinct closed loop among conceptual realizing and visualization.

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Additional resources for A Journey from Robot to Digital Human: Mathematical Principles and Applications with MATLAB Programming

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2 φ3 φ4 = I + φ− + ··· K + − + · · · K2 3! 2! 4! 8) where the well-known Taylor expansions have been applied: sin φ = φ − φ3 φ5 + − ··· 3! 5! and cos φ = 1 − φ2 φ4 + − ···. 2! 4! 8), commonly called the Rodrigues formula, provides a means as a forward direction to determine the equivalent rotation matrix R if a frame is to be rotated by a given angle φ about a given unit vector k. , tr(R) = 3 − 2(1 − cos φ) = 1 + 2 cos φ so that φ = arccos tr(R) − 1 2 . 8) yields R − RT = 2 sin φK, and finally we have K= R − RT .

R − 5 1 − 2 According to Lemma 2, one cannot say at this time that it is a special orthogonal dual matrix until testing its real part R and SRT to see if R ∈ SO(3) and SRT is a skew-symmetric matrix. In fact, 40 2 Mathematical Preliminaries ⎛ 1 RRT = ⎝ 0 0 while ⎛ ⎞⎛ 0 0 1 0 −1 ⎠ ⎝ 0 1 0 0 ⎞ 0 0 0 1⎠ = I −1 0 ⎞⎛ 0 5 7 1 SRT = ⎝ 7 −2 0 ⎠ ⎝ 0 −5 0 −2 0 with det(R) = +1, ⎞ ⎛ ⎞ 0 0 0 −7 5 0 1⎠ = ⎝ 7 0 −2 ⎠ −1 0 −5 2 0 is obviously a skew-symmetric matrix. , R ˆ ∈ SO(3, d). , p = (2 5 7)T . The special orthogonal dual matrices in SO(3, d) Lie group over the dual ring and their transformations will be revisited and further developed in Chapter 4 towards an important and useful theorem for robot kinematic modeling and path-planning applications.

11) to avoid its singularity and do nothing. 11). Nevertheless, in this special case, sin φ = 0 and cos φ = −1. 8), R = I + sin φK + (1 − cos φ)K 2 = I + 2K 2 , or K2 = 1 (R − I). 13) Therefore, you have to use this equation to determine K if R = RT but R = I. For example, if ⎛ ⎞ 0 −1 0 R = ⎝ −1 0 0 ⎠, 0 0 −1 which is symmetric but R = I. Thus, tr(R) = −1 and φ = 1800 . 3 The Exponential Mapping and k–φ Procedure ⎛ 0 K = ⎝ k3 −k2 such that Because ⎛ −k32 − k22 2 K = ⎝ k1 k2 k1 k3 −k3 0 k1 27 ⎞ k2 −k1 ⎠ 0 k1 k2 −k32 − k12 k2 k3 ⎞ k1 k3 k2 k3 ⎠ .

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A Journey from Robot to Digital Human: Mathematical Principles and Applications with MATLAB Programming by Edward Y. L. Gu


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