By Edward Y. L. Gu
This publication presents readers with a high-quality set of diverse and crucial instruments for the theoretical modeling and regulate of complicated robot structures, in addition to for electronic human modeling and reasonable movement iteration. Following a entire advent to the basics of robot kinematics, dynamics and keep an eye on structures layout, the writer extends robot modeling systems and movement algorithms to a far higher-dimensional, greater scale and extra subtle examine region, specifically electronic human modeling. many of the equipment are illustrated via MATLAB codes and pattern graphical visualizations, supplying a distinct 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
Based on the deﬁnition, a 3 by 3 rotation matrix R ∈ SO(3) has the following major properties and applications: 1. R−1 = RT . 2. If a rotation matrix R is also symmetric, then R2 = RRT = I. This means that a vector rotated by such a symmetric R successively twice will return to itself. This also implies that a symmetric rotation matrix acts either a 00 rotation or a 1800 rotation, and nothing but these two cases. 3. One of the three eigenvalues for a rotation matrix R ∈ SO(3) must be +1, and the other two are either both real or complex-conjugate such that their product must be +1.
Let the common normal vector to both the base z0 -axis z0 = (0 0 1)T and the new z1 -axis be determined through the cross-product by ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 1 −1 0 −1 1 b = z0 × z0 = ⎝ −1 0 −1 ⎠ ⎝ 0 ⎠ = ⎝ −1 ⎠ . 1 1 0 1 0 26 2 Mathematical Preliminaries Then, we pick the unit vector of b to be the rotating axis k: ⎛ ⎞ ⎛ ⎞ −1 0 0 −1 1 1 b = √ ⎝ −1 ⎠ such that K = √ ⎝ 0 0 1 ⎠. k= b 2 2 1 −1 0 0 To ﬁnd the rotating angle φ from the base z0 -axis to the new z1 -axis about the above k, we use the dot-product: √ z0T z01 = −1 = z0 z01 cos φ = 3 cos φ, so that 1 cos φ = − √ 3 and sin φ = 1 − cos2 φ = 2 .
To verify the numerical result, we may symbolically calculate f (x) = (1 − 2x2 ) exp(−x2 ). 098 × 10−3 . In the later chapters, we will have to calculate derivatives for much more complicated functions, or high-dimensional vector ﬁelds or matrix ﬁelds. It can be diﬃcult or even unmanageable to ﬁnd their symbolical derivatives by hand before programming them for numerical computation. In this case, the above property owned by the dual ring will overwhelmingly demonstrate its unique advantage over any other algorithms for a numerical derivative solution .