Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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(1.19)
According to Eq. (1.12), θqp = θpq. However, according to the right‐hand rule,
(1.20)
Therefore,
(1.21)
1.4 Reference Frames
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 1.1 A reference frame.
In Eq. (1.22), A is the origin 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
(1.24)
In Eq. (1.24), δij 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
(1.25)
In Eq. (1.25), εijk 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), σijk 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
(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),
(1.31)