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Sequences of rotations

A minimal parameterization of orientation can be obtained by using a set of three angles \(\{\alpha, \beta, \gamma\}\) together with an admissible composition sequence. A sequence of rotations is considered to be "admissible" if no two successive rotations are made around parallel axes. For example, Figure 1 depicts three successive rotations around the \(x \to y \to z\) axes of a fixed frame \(\Sigma_0\) (all at angles \(\pi/2\))

Image title
Figure 1: Example of XYZ rotations around a fixed frame.
  • \(R_x(\pi/2)\)
  • \(R_y(\pi/2)R_x(\pi/2)\)
  • \(R_z(\pi/2)R_y(\pi/2)R_x(\pi/2)\)
Image title
Figure 1: Example of ZYX rotations around the current frame.
  • \(R_z(\pi/2)\)
  • \(R_z(\pi/2)R_y(\pi/2)\)
  • \(R_z(\pi/2)R_y(\pi/2)R_x(\pi/2)\)