Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

      (1.23)

      All the reference frames that are used in this book are selected to be orthonormal, right‐handed, and equally scaled on their axes.

      A reference frame, say , is defined to be orthonormal if its basis vectors are mutually orthogonal and each of them is a unit vector, i.e. a vector normalized to unit magnitude. The orthonormality of can be expressed by the following set of equations that are obeyed by its basis vectors for all i ∈ {1, 2, 3} and j ∈ {1, 2, 3}.

(1.24)

      In Eq. (1.24), δ_ij is defined as the dot product index function, which is also known as the Kronecker delta function of the indices i and j.

      A reference frame, say , is defined to be right‐handed if its basis vectors obey the following set of equations for i ∈ {1, 2, 3}, j ∈ {1, 2, 3}, and k ∈ {1, 2, 3}.

(1.25)

      In Eq. (1.25), ε_ijk is defined as the cross product index function, which is also known as the Levi‐Civita epsilon function of the indices i, j, and k. It is defined as follows:

(1.26)

      Of course, the cross product formula in Eq. (1.25) produces nonzero results only if the indices i, j, and k are all distinct. Therefore, by allowing the indices i, j, and k to assume only distinct values, i.e. by allowing ijk to be only such that ijk ∈ {123, 231, 312; 321, 132, 213}, the considered cross product can also be expressed by the following simpler formula, which does not require a summation operation.

(1.27)

      In Eq. (1.27), σ_ijk is designated as the cross product sign variable, which is defined as follows only for the distinct values of the indices i, j, and k.

      (1.28)

1.5 Representation of a Vector in a Selected Reference Frame

A vector can be resolved in a selected reference frame as shown below.

(1.29)

      Owing to the numerical index notation, Eq. (1.29) can also be written compactly as follows:

(1.30)

      In Eqs. (1.29) and (1.30), is defined as the kth component of in . It is obtained as

      (1.31)

      The components of can be stacked as follows to form a column matrix , which is defined as the column matrix representation of the vector in .

      (1.32)

      In order to show the resolved vector explicitly, may also be denoted as . That is,

      (1.33)

      The basis vector of is represented by the following column matrix in .

(1.34)

      In Eq. (1.34), is the kth basic column matrix, which is defined as shown below for each k ∈ {1, 2, 3}.

      (1.35)

      Here, it must be pointed out that, just like a scalar, is an entity that is not associated with any reference frame. This is because represents in its own frame , whatever is. In other words,

(1.36)

      Moreover, the set of the basic column matrices forms the primary basis of the space of the 3 × 1 column matrices. In other words, any arbitrary column matrix