M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

According to Eq. Set (2.58), . Therefore, Eq. (2.59)becomes

(2.60)

At this point, by using Eq. (2.55), Eq. (2.60)can also be written as

(2.61)

Since is an orthonormal matrix, it obeys Eq. (2.48). That is,

(2.62)

Therefore, Eq. (2.61)leads to the verification that

(2.63)

The rotation matrices have several mathematical properties that turn out to be quite useful especially in the symbolic matrix manipulations required in the analytical treatments within the scope of rotational kinematics. These properties are shown and explained below in Sections 2.7.1and 2.7.2. In both sections, the rotation matrices are expressed in exponential form and the unit vectors of the rotation axes are represented by plain column matrices, such as , without explicit frame indication.

1 Determinant of a Rotation Matrix

As verified in Section 2.6,

(2.64)

1 Inversion of a Rotation Matrix

As also verified in Section 2.6,

(2.65)

Equation (2.65)shows that a rotation can be reversed either by reversing the rotation angle or by reversing the unit vector of the rotation axis.

1 Combination of Successive Rotation Matrices

Two successive rotations about skew axes are neither commutative nor additive. In other words, if ,

(2.66)

Two successive rotations about parallel or coincident axes are both commutative and additive. In other words, if ,

(2.67)

1 Additional Full, Half, and Quarter Rotations

The effect of a full additional rotation is nil. That is,

(2.68)

The effect of a half additional rotation can be expressed as follows:

(2.69)

The effect of a quarter additional rotation can be expressed as follows:

(2.70)

In the preceding formulas, σ is an arbitrary sign variable, i.e. σ = ± 1.

1 Effectivity of a Rotation Operator

A rotation operator is ineffective on the unit vector of its own axis. That is,

(2.71)

However, one must be careful that

(2.72)

1 Angular Differentiation of a Rotation Matrix

A rotation matrix can be differentiated with respect to θ as follows:

(2.73)

1 Rotation About Rotated Axis

Let be rotated into by so that

(2.74)

Then, it can be shown that Eq. (2.74)leads to the following equations.

(2.75)

(2.76)

Equation (2.76)is the expression of the rotation about rotated axis formula .

1 Shifting Formulas for the Rotation Matrices

The following two formulas, which are called shifting formulas , can be obtained as two consequences of Eq. (2.76).

(2.77)