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

DSTO–TN–0640the zenith, the more its azimuthal motion (motion about the vertical axis) constricts itstube to making smaller movements. This means that small movements around the zenithtranslate to large changes in the azimuthal angle, which can be difficult to achieve evenly.Mechanically this constricted movement is called gimbal lock. In using these two anglesto track an aircraft, rapid changes in azimuth can be difficult to model in any numericalalgorithm, and the term gimbal lock has also come to be applied to numerical difficulties.The result is that it has given Euler angles a bad name. But the problem is purely one ofnumerical stability; there’s nothing wrong with using the two Euler angles in principle.This constriction of motion about one axis also causes mechanical problems with somegyroscopes. To picture why this might be, imagine a very simple (and not very useful)example of an inertial navigation system: a horizontal spinning flywheel carried on a yokeattached to hinges at the end of each wingtip, and held underneath the aircraft while itflies straight and level. The flywheel’s axis determines the up/down direction used bythe autopilot to help keep the aircraft on course. The aircraft can pitch up and downand yaw in the local north–east plane without affecting the flywheel’s motion. But if thehuman pilot tries to roll, the flywheel will be forced out of its spin plane. In such an event,the inertial navigation system will be fooled into believing that the aircraft has begunto develop some serious instabilities which it will try to correct, with perhaps disastrousconsequences. While this is a big simplification of a real system, it does encapsulate theproblem that can happen with a real gyroscope when the plane’s attitude reaches someextreme position.One way of avoiding numerical instabilities in tracking algorithms is to use Euler’stheorem, which does away with having to use three angles around the three fixed axes. Soit is that our matrix A in (3.2) describes this rotation. In principle it should be sufficientfor all purposes, although in practice its nine entries can require a powerful ability fornumber crunching, especially when used to describe motions where the orientation anglesare changing with time.3.3 The Workhorse: Rotating in Three DimensionsThe matrix A in (3.2) describes the rotation that changes a body’s orientation, providedwe know how the basis vectors change. But usually we are not given the final positionof those vectors. In general, we need to be able to rotate an arbitrary vector around anarbitrary axis, and this is a slightly different task to be done.This rotation can be done by multiplying the vector by a matrix R n (θ) that we writedown in this section. R n (θ) specifies a right-handed rotation through some angle θ aboutan axis aligned with a unit vector n. (Requiring n to have unit length makes the expressionssimpler.) It rotates the vector x to give x ′ , as shown in Figure 4:Write the unit vector that points along the axis of rotation asx ′ = R n (θ) x. (3.5)⎡ ⎤n 1n = ⎣n 2⎦ , (3.6)n 37