Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

hand, as observed in Eqs. (2.153,2.154,2.155,2.156,2.157,2.158,2.159), σ′ (i.e. the sense of ) is not related to σ = sgn(θ). As a matter of fact, the angle‐axis pair leads to the same rotation matrix, whatever σ and σ′ are. This statement is confirmed as shown below.

(2.161)

Owing to Eq. (2.161), whatever σ is, the sign ambiguity caused by σ′ can again be eliminated by selecting the option with σ′ = + 1, if θ and are required to be determined only once in a while. However, if θ and are required to be determined frequently in a case such that the values of θ happen to be σπ at certain successive instants, then it may be more appropriate to prevent the possibility that changes its direction abruptly as soon as θ becomes σπ. In other words, instead of insisting on the choice σ′ = + 1, it may be preferable to choose σ′ so that with θ(t) = σπ and with are almost codirectional, i.e. .

2.10 Definition and Properties of the Double Argument Arctangent Function

      The double argument arctangent function, which is denoted as atan₂(η, ξ), is defined so that it satisfies the following identity for any ρ > 0.

      (2.162)

      The most characteristic feature of the function atan₂(η, ξ) is that it gives θ without any quadrant ambiguity in the following interval, whenever η ≠ 0 and ξ ≠ 0.

      More specifically, the function atan₂(η, ξ) gives the outcome θ as follows depending on the values of the arguments η and ξ:

3 Matrix Representations of Vectors in Different Reference Frames and the Component Transformation Matrices

      Synopsis

As mentioned in Chapter 1, the vectors are independent of the reference frames in which they are observed. However, their components are naturally dependent on the selected observation frames. In other words, they have different components and matrix representations in different reference frames. Their matrix representations in different reference frames are related to each other by means of the transformation matrices. The transformation matrix between two reference frames can be expressed in various ways. It can be expressed as a rotation matrix, or as a direction cosine matrix, or as a function of the Euler angles of a selected sequence, or as a function of the basis vectors of the relevant reference frames. All these expressions are studied in this chapter together with several examples. Moreover, the matrix representations of the position vectors of a point in differently oriented and/or located reference frames can be related by means of either affine or homogeneous transformations. So, the affine and homogeneous transformations together with the 4 × 4 homogeneous transformation matrices are also studied in this chapter.

3.1 Matrix Representations of a Vector in Different Reference Frames

Consider a vector and two different reference frames and , which are shown in Figure 3.1. Both and are assumed to be orthonormal, right‐handed, and equally scaled on their axes. They have different orientations described by the following basis vector triads.

      (3.1)

      In and , the observed vector is resolved differently as shown below.

(3.2)

In Eq. (3.2), the components of in and are obtained as follows for k ∈ {1, 2, 3}:

      (3.3)

      (3.4)