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

DSTO–TN–0640so that (3.2) yields R y (90 ◦ ) trivially.) The required rotated vector is thenas expected.⎡ ⎤ ⎡ ⎤⎡⎤ ⎡ ⎤2 0 0 1 2 0R y (90 ◦ ) ⎣0⎦ = ⎣ 0 1 0⎦⎣0⎦ = ⎣ 0⎦ , (3.12)0 −1 0 0 0 −2An example of combining two rotations is given in Appendix A. As stated by Euler’stheorem, the effect of these is just one rotation, although the values of the resultingequivalent axis and angle turned through are not obvious at all from the original two axesand angles.The rows of a rotation matrix will gradually lose their mutual orthonormality becauseof rounding errors under repeated iterations in a computer programme. This wanderingcan be kept in check by periodically re-orthogonalising, accomplished byR → R ( R t R ) −1/2 . (3.13)Rotation Order and its Possible Blind ApplicationSince matrix multiplication is not commutative, two rotations are also not in general commutative:the result depends on the order in which they are done. However, infinitesimalrotations are commutative. This fact can be used to our advantage when writing computercode that takes inputs from a joystick, in order to rotate a scene (say, in a flightsimulator). The pilot might induce both a pitch and a roll, but only by applying each ofthese in tiny increments will the software successfully reproduce the effect that the pilotwants.Nevertheless, it’s possible to go to an extreme of demanding a set rotation order thatdoes not mimic what the user of a software package wants, but that does do somethinghighly misleading. In fact, a search through the internet shows that this mistake seems tobe a classic of computer graphics programming, and has caused some in that community todistrust the mathematics of rotation. An example of this incorrect application of rotationsis discussed in Appendix C.3.4 Rotating using QuaternionsAlthough the rotation matrix R n (θ) has nine entries, it’s really only built from three numbers:the angle θ turned through, and any two components of n; the third component of nis then implied, since n has unit length. Using R n (θ) can be computationally inefficient,since not only do all nine components need to be manipulated, but if the matrix is appliedwithin a loop, perhaps thousands of times, numerical inaccuracies can degrade itsorthogonality. That is, its rows (or equivalently columns) will slowly lose their mutualorthonormality. It can certainly be periodically re-orthogonalised after an appropriatenumber of iterations, as shown in (3.13). Nevertheless, the question arises as to whetherany way exists of rendering the matrix down to its basic three numbers, and perhapseliminating some of the numerical complexity in the process.9