Define t = 0 to be the exact moment of the summer solstice, and the global reference frame to be coincident with the Earth’s frame at time t = 0. Give an expression R(t) for the rotation matrix that represents the instantaneous orientation of the earth at time t. Determine as a function of time the homogeneous transformation that specifies the Earth’s frame with respect to the global reference frame.42 In general, multiplication of homogeneous transformation matrices is not commutative. Consider the matrix product Determine which pairs of the four matrices on the right-hand side commute. Explain why these pairs commute. Find all permutations of these four matrices that yield the same homogeneous transformation matrix, .
Figure 2.13 Diagram for Problem 2–37.
Figure 2.14 Diagram for Problem 2–38.
Notes and References
Rigid body motions and the groups SO(n) and SE(n) are often addressed in mathematics books on the topic of linear algebra. Standard texts for this material include [9], [30], and [49]. These topics are also often covered in applied mathematics texts for physics and engineering, such as [143], [155], and [182]. In addition to these, a detailed treatment of rigid body motion developed with the aid of exponential coordinates and Lie groups is given in [118].
Notes
1 1 We will use , to denote both coordinate axes and unit vectors along the coordinate axes depending on the context.
2 2 It should be noted that other conventions exist for naming the roll, pitch, and yaw angles.
3 3 The definition of rigid motion is sometimes broadened to include reflections, which correspond to detR = −1. We will always assume in this text that detR = +1 so that R ∈ SO(3).
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