Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

(3.43) gives c₂₂ as

      (3.46)

      Finally, the third column is found as follows according to Eq. (3.34):

      (3.47)

      Note that the procedure described above provides four different outcomes for due to the independent sign variables σ₁ and σ₂. To pick up one of these solutions, let σ₁ = σ₂ = + 1. This particular choice of σ₁ and σ₂ leads to , which is shown below.

      (3.48)

      As a check for the validity of the above solution, it can be shown that .

3.4 Expression of a Transformation Matrix as a Direction Cosine Matrix

      3.4.1 Definitions of Direction Angles and Direction Cosines

The rotational deviation between two reference frames, e.g. and , can be represented by the direction angles as shown in Figure 3.2. In that figure, only six of the nine direction angles are illustrated for the sake of neatness. The direction angles between and are denoted and defined as follows for all i ∈ {1, 2, 3} and j ∈ {1, 2, 3}:

      (3.49)

Figure 3.2 Direction angles between two reference frames.

Without any loss of generality, the direction angles can be defined to be positive angles that are confined to the range [0, π]. That is,

      In a direct association with the direction angles, the direction cosines between and are denoted and defined as follows:

      (3.50)

      3.4.2 Transformation Matrix Formed as a Direction Cosine Matrix

      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.

3.5 Expression of a Transformation Matrix as a Rotation Matrix

      3.5.1 Correlation Between the Rotation and Transformation Matrices

      Since the reference frames and are both orthonormal, right‐handed, and equally scaled on their axes, it can be imagined that is obtained by rotating as indicated below.

      (3.54)