Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

imply that is exponentiated as follows:

      (1.64)

      (1.65)

       The ssm operation is applied as shown below on the products and .

      (1.66)

(1.67)

      Here, it is to be noted that Eq. (1.67) is valid if is a rotation matrix, i.e. an orthonormal matrix with .

       It happens that is a singular matrix and its rank is two. That is,

(1.68)

(1.69)

1.8 Examples Involving Skew Symmetric Matrices

      1.8.1 Example 1.1

      This example is about the expansion of the following triple vector product.

(1.70)

      In a selected reference frame , the matrix version of Eq. (1.70) can be written as

(1.71)

      In Eq. (1.71), all the matrices are expressed in the same frame . Therefore, the frame indicating superscript (a) is concealed for the sake of brevity. By expanding the product according to Eq. (1.61)Eq. 1.71 can be written again as follows:

      (1.72)

      The corresponding vector equation written below turns out to be the required expansion of the triple vector product.

      (1.73)

      1.8.2 Example 1.2

As a typical problem involving the singularity of , consider the following equation, from which is to be found for given and .

(1.74)

      Due to Eqs. (1.68) and (1.69), cannot be found uniquely from Eq. (1.74). However, it can be found with the following expression that contains an arbitrary parameter λ.

(1.75)

      In Eq. (1.75), is the part of that is orthogonal to . So, it can be expressed as

      (1.76)

      The coefficient γ is to be determined so as to satisfy Eq. (1.74). That is,

      Since and are orthogonal, . Therefore, Eq. (1.77) gives γ as

      (1.78)

      Hence, and are obtained as shown below.

      (1.79)

      (1.80)

      1.8.3 Example 1.3

      Consider the following 3 × 3 matrix equation, which is to be solved for .

(1.81)

      The matrix can be expressed as follows in terms of its columns.