Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

(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 is observed in three differently oriented reference frames, such as , , and , the following equations can be written to relate its column matrix representations.

(3.21)

(3.22)

(3.23)

On the other hand, the combination of Eqs. (3.21) and (3.22) results in

(3.24)

Equations (3.23) and (3.24) imply that

(3.25)

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 and . The kth basis vector of can be represented by the following column matrix in for k ∈ {1, 2, 3}.

      (3.27)

      Using the transformation matrix between and , can be expressed as follows:

(3.28)

Note that picks up the kth column of the matrix, by which it is postmultiplied. Thus, in Eq. (3.28), happens to be the kth column of . Therefore, can be expressed column by column as shown below.

(3.29)

      3.3.2 Row‐by‐Row Expression

      Alternatively, Eq. (3.28) can also be written as follows by interchanging a and b:

(3.30)

Recalling that , Eq. (3.30) leads to the following equation when it is transposed on both sides.

(3.31)

Note that picks up the kth row of the matrix, by which it is premultiplied. Thus, in Eq. (3.31), happens to be the kth row of . Therefore, as an alternative to Eq. (3.29), can also be expressed row by row as shown below.

      (3.32)

      Here, it is worth paying attention that the column‐by‐column expression of requires the column matrix expressions of the basis vectors of in