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Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
Читать онлайн.Название Kinematics of General Spatial Mechanical Systems
Год выпуска 0
isbn 9781119195764
Автор произведения M. Kemal Ozgoren
Жанр Математика
Издательство John Wiley & Sons Limited
2.6 Orthonormality of the Rotation Matrices
Suppose a vector
(2.46)
Since a rotation operator does not change the magnitude of the vector it rotates, the following equations can be written.
Equation (2.47) implies that
Hence, Eq. (2.48) implies further that
Here, as a reminder from the matrix algebra, a matrix
Owing to its orthonormality, the inverse of
Note that
Referring to Eq. (2.26), Eq. (2.51) implies the following exponential expression.
Equations (2.51) and (2.52) show that the inverse of
Equation (2.53) can be verified as explained below.
The matrix
(2.54)
Since the kth basic column matrix
According to Eq. (2.55),
Since
Here, it must be pointed out that a rotation operator does not only retain the magnitudes of the vectors it rotates but it also retains the right‐hand rule for their cross products. Therefore, the vectors