Three dimensional rotations B

Page 1

1 Three-Dimensional Rotations A rotation is a transformation that preserves the length, the volume (with sign), and the angle between pairs of vectors. If vector a is rotated into vectorb, then the property of length preservation is expressed as |a| = |b|. (1.1) Furthermore, if vectors a1,a2 anda3 are transformed by the same identical rotation into vectors b1, b2 and b3, respectively, by the property of volume preservation we have a1 ·a2 ×a3 =b1 ·b2 ×b3. (1.2) Finally, by the property of angle preservation between vectors, we have cos ϕa1,a2 = a1 ·a2 |a1||a2| = b1 ·b2 |b1||b2| = cos ϕb1,b2 , (1.3) where ϕi,j indicates the angle between vector i and vectorj. Because of these three properties, a rotation is termed a rigid transformation . In mechanics, rotations are used for expressing relationships between co- ordinate systems. A rectangular coordinate system is defined by a frame. Each frame is composed by a point, called the frame pole, and by a triad of orthogonal unit vectors with origin in the pole. Since rotations are rigid transformations, given two triads I and J , the latter can always be seen as obtained by a suitable rotation of the former. Hence, rotations play a funda- mental role whenever a problem is formulated in terms of multiple coordinate systems in relative motion with respect to one another. 1 2 THREE-DIMENSIONAL ROTATIONS In the formulation of flight mechanics problems, rotations are routinely used for transforming the components of vectors from one coordinate system to another. Furthermore, the motion of a vehicle with respect to a frame of reference is described in terms of the position and orientation of a body attached frame that is fixed to the vehicle and moves with it. The position of the vehicle with respect to the reference frame is given by…