Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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Equation (3.17) implies the following successive equations.
(3.18)
Furthermore, when Eqs. (3.19) and (3.14) are compared, it is seen that
(3.20)
As seen above, the inverse of a transformation matrix is equal to its transpose. This property makes a transformation matrix an element of the set of orthonormal matrices just like a rotation matrix. The orthonormality of a rotation matrix was shown in Chapter 2.
1 (c) Combination Property
If a vector
On the other hand, the combination of Eqs. (3.21) and (3.22) results in
Equations (3.23) and (3.24) imply that
Equation (3.25) can be extended to more than three reference frames as follows:
(3.26)
3.3 Expression of a Transformation Matrix in Terms of Basis Vectors
3.3.1 Column‐by‐Column Expression
Consider two reference frames
(3.27)
Using the transformation matrix between
Note that
3.3.2 Row‐by‐Row Expression
Alternatively, Eq. (3.28) can also be written as follows by interchanging a and b:
Recalling that
Note that