Download e-book for kindle: A Journey from Robot to Digital Human: Mathematical by Edward Y. L. Gu

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By Edward Y. L. Gu

ISBN-10: 3642390463

ISBN-13: 9783642390463

This booklet presents readers with an outstanding set of varied and crucial instruments for the theoretical modeling and regulate of advanced robot platforms, in addition to for electronic human modeling and practical movement new release. Following a complete advent to the basics of robot kinematics, dynamics and keep an eye on platforms layout, the writer extends robot modeling approaches and movement algorithms to a far higher-dimensional, better scale and extra subtle study zone, specifically electronic human modeling. lots of the tools are illustrated via MATLAB codes and pattern graphical visualizations, supplying a distinct closed loop among conceptual figuring out and visualization.

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

Example text

R + S ∈ SO(3, d), SRT must be a skewBased on Lemma 2, for a R symmetric matrix so that RS T = −SRT or RS T + SRT = O. , P = p×. 23), then the orientation of the frame is uniquely described by the real part R and the position of the frame origin is determined by the cross-product operator P = SRT . For example, given a 3 by 3 dual matrix ⎛ ⎞ 1 5 7 ˆ = ⎝ 7 − 2 −1 ⎠ = R + S. 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.

Let x1 , · · · , xm be m vectors in Rn space (m ≤ n). For any i, j = 1, · · · , m, xi ∧ xj is called a wedge or exterior product if the following conditions hold: 1. Skew-symmetric: xi ∧ xj = −xj ∧ xi ; 2. Distributive: (a1 xi + a2 xj ) ∧ xk = a1 xi ∧ xk + a2 xj ∧ xk , for two arbitrary scalar constants a1 and a2 ; and 3. Associative: xi ∧ (xj ∧ xk ) = (xi ∧ xj ) ∧ xk . The first condition of skew-symmetry in the definition implies that for the same vector, xi ∧ xi = 0 always. Without loss of generality, let x1 , x2 and x3 be three independent 3D vectors in R3 .

2 ⎞⎛ ⎞ ⎛ ⎞ 0 −1 0 −3 0 −2 ⎠ ⎝ 1 ⎠ = ⎝ −6 ⎠ . 2 0 3 2 The triple scalar product becomes ⎛ ⎞ −3 (v3 v1 v2 ) = v3 · (v1 × v2 ) = (−4 0 2) ⎝ −6 ⎠ = 16. 2 On the other hand, ⎞ ⎛ −4 0 v3T det ⎝ v1T ⎠ = det ⎝ 2 −1 v2T 0 1 ⎛ ⎞ 2 0 ⎠ = 16. 3 Thus, both approaches to evaluating the triple scalar product agree with each other. Furthermore, the triple vector product can be calculated by ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 −2 0 −3 12 v3 × (v1 × v2 ) = S(v3 )(v1 × v2 ) = ⎝ 2 0 4 ⎠ ⎝ −6 ⎠ = ⎝ 2 ⎠ . 0 −4 0 2 24 It can also be evaluated by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 0 12 v3 × (v1 × v2 ) = (v3T v2 )v1 − (v3T v1 )v2 = 6 ⎝ −1 ⎠ + 8 ⎝ 1 ⎠ = ⎝ 2 ⎠ .

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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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