Читать книгу Robot Modeling and Control - Mark W. Spong - Страница 68

Given the skew-symmetric matrix S(k) it is easy to show that S³(k) = −S(k), from which it follows that S⁴(k) = −S²(k), etc. Thus the series expansion for e^S(k)θ reduces to

the latter equality following from the series expansion of the sine and cosine functions. The expression

(2.57)

is known as Rodrigues’ formula. It can be shown by direct calculation that the angle-axis representation for R_{k, θ} given by Equation (2.44) and Rodrigues’ formula in Equation (2.57) are identical.

Remark 2.1.

The above results show that the matrix exponential function defines a one-to-one mapping from so(3) onto SO(3). Mathematically, so(3) is a Lie algebra and SO(3) is a Lie group.