Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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The fact that
When c22 is eliminated between Eqs. (3.42) and (3.43), the following quadratic equation is obtained for c32.
(3.44)
Hence, c32 is found as follows with an additional sign ambiguity represented by σ2 = ± 1:
(3.45)
Upon inserting c32, Eq. (3.43) gives c22 as
(3.46)
Finally, the third column
(3.47)
Note that the procedure described above provides four different outcomes for
(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.
(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
(3.50)
3.4.2 Transformation Matrix Formed as a Direction Cosine Matrix
Since the basis vectors of
Using the transformation matrix
As mentioned before,