Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 58

Since the basis vectors of and are unit vectors, the direction cosines can also be defined by the following dot product equation written for all i ∈ {1, 2, 3} and j ∈ {1, 2, 3}.

(3.51)

Using the transformation matrix and the matrix representations of and in one of the reference frames and , say , Eq. (3.51) can also be written and manipulated as shown below.

(3.52)

As mentioned before, and pick up the ith row and jth column of the matrix they multiply. Therefore, happens to be the i‐j element of according to Eq. (3.52). Owing to this fact, can be constructed as a direction cosine matrix, i.e. as a matrix constructed as follows by stacking the direction cosines between and .

(3.53)

In Eq. (3.53), cθ is used as an abbreviation for cosθ.