Using Rotations to Build Aerospace Coordinate Systems - Defence ...

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DSTO–TN–0640But the effect of this latest rotation is not what we might have expected. It is reasonableto believe, based on trying each slider in turn on its own, that each slider rotates about itsrespective axis. When we increased the x-slider from 10 ◦ to 15 ◦ , most users would expectthe object to rotate by a further 5 ◦ about x. But this is not what happens. The softwaredoes what it was designed to do: it rotates the base orientation usingR y (10 ◦ ) R x (15 ◦ ).(C1)The rotation suggested by our intuition is an increment of 5 ◦ about x on the last orientation—which was represented by R y (10 ◦ ) R x (10 ◦ ) acting on the base orientation—orR x (5 ◦ ) R y (10 ◦ ) R x (10 ◦ ).(C2)Expressions (C1) and (C2) are not the same. In fact (C1) could have been writtenR y (10 ◦ ) R x (5 ◦ ) R x (10 ◦ ),(C3)which differs from (C2) in that two rotations are swapped. Because rotation is noncommutative,these two different matrices (C2) and (C3) produce different orientations whenmultiplying the base orientation. To reiterate, our intuition expected that by making threeslider adjustments, first incrementing the x-slider (from zero) by 10 ◦ , then incrementingthe y-slider (from zero) by 10 ◦ , then incrementing the x-slider by 5 ◦ , that (C2) wouldresult. In fact, (C1) or equivalently (C3) resulted. That might be unintuitive, but themathematics and the computer software have done exactly what was asked of them.Our confusion over the effects of the sliders seems to reach its worst if the y-slider isfirst set to rotate through −90 ◦ , and then the x- and z-sliders are changed arbitrarily.Because the following identity holds for all angles α, β,⎡⎤0 − sin(α + β) − cos(α + β)R z (α) R y (−90 ◦ ) R x (β) = ⎣0 cos(α + β) − sin(α + β) ⎦ ,1 0 0(C4)the total rotation—being a function only of the sum of the x- and z-slider angles—doesnot distinguish between the x- and z-sliders. Thus the x- and z-sliders have the sameeffect, making it appear that we have lost one rotation axis somewhere, as if, to use anoft-quoted phrase, “the x-axis has been rotated onto the z-axis” (!) But axes are not seton hinges; there is no such thing as a flexible axis, and axes cannot be rotated onto eachother. The software is doing exactly what it was designed to do. Whether our intuitionwould prefer a different behaviour is another matter entirely.This apparent loss of one axis is erroneously labelled gimbal lock by some graphicsprogrammers, who confuse it with the real numerical Euler angle problem described earlieron page 7. But there is no such problem with Euler angles in software rotations such asdescribed above, and this use of the term is a misnomer. Unfortunately, historically thesemisinterpretations went hand in hand with using matrices to perform rotations, with theresult that matrices and Euler angles are sometimes seen as not up to the task. This isdefinitely wrong; both matrices and Euler angles can handle any rotation asked of them.26
DSTO–TN–0640Restoring an Intuitive Feel to the Sliders If we wish the sliders to act in an intuitiveway, then each time we adjust any slider, we must not have the software read the threeslider values and rotate the base orientation through them all in some preset order. Rather,if we adjust the x-slider by 2 ◦ , the software should read the values, note what has changed,and rotate the current orientation about the x-axis through the increment of 2 ◦ . Notonly does this make the sliders and rotations behave as our intuition suggests, but it alsomakes for faster processing, since only one rotation is ever performed per slider interaction,instead of three.27
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Page 12 and 13: DSTO-TN-0640ZmeridianPrimeYXEquator
Page 14 and 15: DSTO-TN-0640ZNorthEastωDownαYXFig
Page 16 and 17: DSTO-TN-0640then suppose they are t
Page 18 and 19: DSTO-TN-0640nθx ′xFigure 4: The
Page 20 and 21: DSTO-TN-0640We are always free to c
Page 22 and 23: DSTO-TN-0640ZNDEN 0E 0D 0 N 1E 1ω
Page 24 and 25: DSTO-TN-0640The vector giving the p
Page 26 and 27: DSTO-TN-0640Equation (2.2) together
Page 28 and 29: DSTO-TN-0640Next, θ is the angle b
Page 30 and 31: DSTO-TN-06405 Concluding RemarksAll
Page 32 and 33: DSTO-TN-0640Using QuaternionsThe ne
Page 34 and 35: DSTO-TN-0640xyz Angle-Axis to Euler
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Page 40 and 41: DSTO-TN-0640Way point(0.9, 0.1, 0.1
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Page 44 and 45: Robert Anderson, WSD 1David Marlow,
Page 46 and 47: Head of Aerospace and Mechanical En

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