By Edward Y. L. Gu
This e-book offers readers with a pretty good set of diverse and crucial instruments for the theoretical modeling and keep watch over of advanced robot structures, in addition to for electronic human modeling and practical movement iteration. Following a finished advent to the basics of robot kinematics, dynamics and keep watch over structures layout, the writer extends robot modeling tactics and movement algorithms to a far higher-dimensional, greater scale and extra refined study region, specifically electronic human modeling. many of the equipment are illustrated through MATLAB codes and pattern graphical visualizations, delivering a distinct closed loop among conceptual realizing and visualization.
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Additional info for A Journey from Robot to Digital Human: Mathematical Principles and Applications with MATLAB Programming
By using the skew-symmetric matrix for v2 , it can be re-written as (v1 v2 v3 ) = v1T S(v2 )v3 . If taking the transpose, then, (v1 v2 v3 ) = −v3T S(v2 )v1 = v3T S(v1 )v2 = v3 · (v1 × v2 ) = (v3 v1 v2 ). Similarly, (v1 v2 v3 ) = (v3 v1 v2 ) = (v2 v3 v1 ) = −(v1 v3 v2 ) = −(v2 v1 v3 ) = −(v3 v2 v1 ). Furthermore, if each vi = (ai bi ci )T for i = 1, 2, 3, then ⎛ ⎞ a1 b 1 c1 (v1 v2 v3 ) = det ⎝ a2 b2 c2 ⎠ . a3 b 3 c3 Therefore, the three vectors lie on a common plane if and only if (v1 v2 v3 ) = 0.
The cross-product operator of a × b. , S(a) has been ⎞ 0 1⎠. 3 The Exponential Mapping and k–φ Procedure ⎛ 2 = ⎝0 6 23 ⎞ ⎛ ⎞ ⎛ ⎞ −2 1 2 0 6 0 −2 −5 5 0 ⎠ − ⎝ −2 5 4 ⎠ = ⎝ 2 0 −4 ⎠ , 4 3 1 0 3 5 4 0 which exactly meets S(c) = c× = S(a × b). In mathematical history, one of the most signiﬁcant discoveries in the theory of Lie group and Lie algebra is the following exponential mapping: EXP : so(3) −→ SO(3). 6) This mapping means that for each 3 by 3 skew-symmetric matrix S ∈ so(3), its exponential function exp(S) = R ∈ SO(3) is always a rotation matrix.
This is just the negative of the vector from the above example, and it does not aﬀect the ﬁnal rotation at all because the rotation angle φ can be chosen by ±1800 in this special case. 7). One will ﬁnd later that the k– φ procedure is extremely useful in robotic kinematics and digital human realistic motion-planning, especially for the orientation path-planning. 4 The Dual Number, Dual Vector and Their Algebras In mathematics, two real numbers a and b can be combined under a certain unit quantity j to create a new number c = a + jb.
A Journey from Robot to Digital Human: Mathematical Principles and Applications with MATLAB Programming by Edward Y. L. Gu