M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(1.18)

If the order of and is reversed, Eq. (1.15)becomes

(1.19)

According to Eq. (1.12), θ qp= θ pq. However, according to the right‐hand rule,

(1.20)

Therefore, . This verifies the well‐known characteristic feature of the cross product that its outcome changes sign when the order of its multiplicands is reversed. That is,

(1.21)

In the three‐dimensional Euclidean space, a reference frame is defined as an entity that consists of an origin and three distinct noncoplanar axes emanating from the origin. The origin is a specified point and the axes have specified orientations. More specifically, the axes of a reference frame are called its coordinate axes . For the sake of verbal brevity, a reference frame may sometimes be called simply a frame . A reference frame, such as the one shown in Figure 1.1, may be denoted in one of the following ways, which convey different amounts of information about its specific features.

(1.22)

Figure 1.1A reference frame.

In Eq. (1.22), A is the origin of . The origin of may also be denoted as O a. The coordinate axes of are oriented so that each of them is aligned with one member of the following set of three vectors, which is denoted as and defined as the basis vector triad of .

(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 ijis 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 ijkis 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 ijkis 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)

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