By Edward Y. L. Gu
This e-book offers readers with a superior set of various and crucial instruments for the theoretical modeling and keep watch over of complicated robot platforms, in addition to for electronic human modeling and real looking movement iteration. Following a finished creation to the basics of robot kinematics, dynamics and keep watch over platforms layout, the writer extends robot modeling strategies and movement algorithms to a far higher-dimensional, better scale and extra subtle learn quarter, particularly electronic human modeling. lots of the equipment are illustrated via MATLAB codes and pattern graphical visualizations, supplying a special 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
Let a ˆ = a + b and cˆ = c + d with each of a, b, c and d to be a 3 by 1 vector. Then, the inner (dot) product between a ˆ and cˆ becomes a ˆT cˆ = aT c + (bT c + aT d). To perform the cross (vector) product a ˆ × cˆ, let the skew-symmetric matrix of a ˆ be deﬁned as 36 2 Mathematical Preliminaries ⎛ 0 S(ˆ a) = ⎝ a3 + b 3 −a2 − b2 −a3 − b3 0 a1 + b 1 ⎞ a2 + b 2 −a1 − b1 ⎠ = S(a) + S(b). 0 Then, a ˆ × cˆ = S(ˆ a)ˆ c = S(a)c + (S(b)c + S(a)d) = a × c + (b × c + a × d). Likewise, the product between two dual matrices Aˆ = A + B and Cˆ = C + D for four n by n square matrices A, B, C and D is similar to the dual vector products: AˆCˆ = AC + (BC + AD), and also ˆa = Aa + (Ab + Ba) Aˆ without exception.
0000 >> As you can see clearly, once the internal function eig(R) is called, the element “val” prints the three eigenvalues along the diagonal of the output matrix, while “vec” prints the three column eigenvectors for the input matrix R. 12). 4 The Dual Number, Dual Vector and Their Algebras 29 This shows that all the three eigenvalues of the 3 by 3 identity I are +1 and their corresponding eigenvectors are given by the unit vectors of the x, y and z axes of the reference frame. 7071 0)T . 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.
At the lower-left corner are always zeros, this ensures that the determinant n det(eA ) = det(P −1 eA P ) = eλi = exp(λ1 + · · · + λn ) = etrA . i=1 Now, applying the Taylor expansion on the exponential function of any dual square matrix M yields exp( M ) = I + M. This implies that if A is nonsingular, exp( A−1 B) = I + A−1 B so that ˆ = det(A + B) = det(A) · det(I + A−1 B) = det(A) · det(e det(A) A−1 B ). 20), we ﬁnally reach a new property for the determinant of a dual square matrix Aˆ as follows: ˆ = det(A + B) = det(A) · e tr(A−1 B) = det(A) · (1 + tr(A−1 B)).