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

DSTO–TN–0640We are always free to construct any quantity from those three numbers. For instance wecould specify a rotation through θ around (n 1 , n 2 , n 3 ) by a new object written as (θ, n 1 , n 2 ).This is very concise, but the catch is that we would also have to define how this objectacts on a vector to rotate it. The result is not anything simple, so we gain nothing bydoing this.But all is not lost. If we allow a little more complexity, and construct a new object Q n (θ)from θ, n 1 , n 2 , n 3 along with a sine and cosine, then this turns out to be simple enough tooffset the more complicated way it has to interact with vectors. Here n is taken to be theunit row vector (n 1 , n 2 , n 3 ):(Q n (θ) ≡ cos θ 2 , nsin θ ). (3.14)2This new object Q n (θ) is called a rotation quaternion. More general quaternions (not forrotating) are composed of a number and a vector: (a 0 , a). Two quaternions multiply inthe following way:(a 0 , a)(b 0 , b) ≡ (a 0 b 0 − a·b, a 0 b + b 0 a + a×b) , (3.15)so that the squared length of a quaternion is defined to be (in analogy to the length of avector):∣∣(a 0 , a) ∣ 2 ≡ a 2 0 + |a| 2 . (3.16)We can see straight away that a rotation quaternion Q n (θ) has unit length, which is theequivalent property of rotation quaternions as orthogonality and a unit determinant is ofrotation matrices.Using rotation quaternions, a row vector x can be rotated through an angle θ arounda unit vector n by applying a double multiplication:(0, x ′ ) = Q n (θ) (0, x)Q −n (θ), (3.17)where quaternion multiplication is associative, so either product can be done first. Asan example, in (3.12) we used a matrix to rotate (2, 0, 0) by 90 ◦ about the y-axis togive (0, 0, −2). Contrast the matrix approach with the quaternion approach:(0, x ′ ) =( )1 1 √2 , √ (0, 1, 0) 2(0, 2, 0, 0)= (1, 0, 1, 0)(0, 1, 0, 0)(1, 0, −1, 0)so that the result is (0, 0, −2) as expected.( 1 √2 , √ −1 )(0, 1, 0) 2= (0, 0, 0, −2), (3.18)Combining two rotations using quaternions is straightforward. Just as a matrix rotationR 1 followed by R 2 is equivalent to a single matrix rotation R 2 R 1 , a quaternionrotation Q 1 followed by Q 2 is equivalent to a single quaternion rotation Q 2 Q 1 .Finally, just as a rotation matrix will slowly lose orthogonality under repeated iterationsin a software routine, so a rotation quaternion’s length will slowly wander from one in thesame circumstances. It can be periodically reset by dividing each of the quaternion’s fourelements by the length calculated from (3.16), which is a very much simpler task thanre-orthogonalising a matrix in (3.13).10