FIGURE 6 Matrix logarithm and matrix exponential based decomposition of the rotational component of the numerical example illustrating three-dimensional rigid-body movements. See Fig.4 legend for additional details. In this case, rotation is applied around the fixed axis. Note that the positions of the two round dots do not change, and that the intervening positions constitute optimal intermediates without extraneous movement. The decomposition into three steps is arbitrary,the required 18.8571° rotation can be equally divided among any number of intervening positions. If the translational component of the numerical example had been included, the rotations in the fourth row would be centered around the (x, y) point ( — 2.8227, — 0.1917). The two points on the "fixed axis" and the origin would rotate around this point in the images on the fourth row, and all points would be translated perpendicular to the plane of the page in the fourth row by 2.0314 units with each of the three transformation steps. See also Plate 67.

FIGURE 6 Matrix logarithm and matrix exponential based decomposition of the rotational component of the numerical example illustrating three-dimensional rigid-body movements. See Fig.4 legend for additional details. In this case, rotation is applied around the fixed axis. Note that the positions of the two round dots do not change, and that the intervening positions constitute optimal intermediates without extraneous movement. The decomposition into three steps is arbitrary,the required 18.8571° rotation can be equally divided among any number of intervening positions. If the translational component of the numerical example had been included, the rotations in the fourth row would be centered around the (x, y) point ( — 2.8227, — 0.1917). The two points on the "fixed axis" and the origin would rotate around this point in the images on the fourth row, and all points would be translated perpendicular to the plane of the page in the fourth row by 2.0314 units with each of the three transformation steps. See also Plate 67.

an instantaneous elementary transformation. Conversion of the rotational components from radians to degrees anticipates the results of computing the corresponding matrix exponentials to derive the instantaneous elementary transformations.

The instantaneous elementary rotational transformation around the x-axis is

x'" |
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