Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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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 , whereas the row‐by‐row expression of the same requires the row matrix expressions of the basis vectors of in .

      3.3.3 Remark 3.1

      Equation (3.29) shows that the columns of a transformation matrix, e.g. , represent the members an orthonormal vector triad, i.e. . This fact poses six independent scalar constraint equations on the nine elements of . The independent constraint equations can be expressed as follows for i ∈ {1, 2, 3} and j ∈ {1, 2, 3}:

      (3.33)

      Therefore, can be expressed completely in terms of only three independent parameters.

      3.3.4 Remark 3.2

      The basis vector triad associated with is not only orthonormal but also right‐handed. Therefore, one of the columns of can be obtained from its other two columns by using the cross product operation. For example, and hence can be obtained as follows by using the matrix equivalent of the cross product operation:

(3.34)

Note that Eq. (3.34) gives three elements of readily in terms of its other six elements. In other words, it already provides three independent constraint equations on the nine elements of . Therefore, in the presence of Eq. (3.34), only three additional independent constraint equations can be posed on the elements of . These three additional constraint equations are written as follows involving the first two columns of :

      (3.35)

      (3.36)

      (3.37)

      3.3.5 Remark 3.3

      Remarks 3.1 and 3.2 can also be stated similarly for the rows of .

      3.3.6 Example 3.1

      In this example, the three independent parameters of , which is briefly denoted here as , are selected as its three elements, which are c₁₁, c₂₁, and c₁₂. The following values are specified for them.

      (3.38)

      It is required to determine by finding its remaining six elements.

      The column‐by‐column expression of can be written as

      (3.39)

      By the given information, the first two columns of are determined partially as shown below.

      (3.40)

      Since , c₃₁ is found as follows with a sign ambiguity represented by σ₁ = ± 1:

      (3.41)

      The fact that leads to the following equation.

(3.42)

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