M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

(1.78)

Hence, and are obtained as shown below.

(1.79)

(1.80)

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.

(1.82)

Along with , can be expressed as follows in terms of its elements.

(1.83)

Hence, Eq. (1.81)can be written in a more detailed form as

(1.84)

Equation (1.84)leads to the following scalar equations with the indicated premultiplications.

(1.85)

(1.86)

(1.87)

Note that, for i ∈ {1, 2, 3} and j ∈ {1, 2, 3},

(1.88)

Thus, Eqs. (1.85)– (1.87)reduce to the following equations.

(1.89)

(1.90)

(1.91)

Equations (1.89)– (1.91)imply that

(1.92)

Therefore, if , Eqs. (1.89)– (1.91)give the elements of as follows:

(1.93)

(1.94)

(1.95)

Note that the solution obtained above is the same as the solution provided by Cramer's rule .

This chapter is devoted to the rotation of vectors and the rotation operators that rotate vectors. The rotation of a vector is expressed both as a vector equation and as a matrix equation written in a selected reference frame. The vector equation is obtained as the Rodrigues formula. The matrix equation is written in terms of the rotation matrix, which is the matrix representation of the rotation operator in the selected reference frame. The expression of the rotation matrix is obtained in terms of the angle of rotation and the unit vector along the axis of rotation. It is shown that the rotation matrix can be expressed very compactly in the exponential form. This chapter also presents the salient mathematical properties of the rotation matrices that can be used conveniently in the symbolic manipulations concerning rotational kinematics. Demonstrative examples are also included.

Figure 2.1illustrates the rotation of a vector into another vector by an angle θ about an axis described by a unit vector .

Incidentally, a vector may be acted upon, simultaneously or successively, by two kinds of displacement operators . One of them is a rotation operator , which is defined as an operator that changes only the orientation of a vector irrespective of any possible change in its location. The other one is a translation operator , which is defined as an operator that changes only the location of a vector without changing its orientation.

As mentioned above, a rotation operator is not affected by any translational displacement. Therefore, without any loss of generality, the rotation of is illustrated in Figure 2.1so that the point O (i.e. the tail point of ) is assumed to be fixed and the rotation axis is assumed to pass through that point. Moreover, as illustrated on the left‐hand side of Figure 2.1, the vector moves on the surface of a cone while it is rotated into the vector . The projected appearance of this rotation on the base of the mentioned cone is illustrated on the right‐hand side of Figure 2.1.