Three dimensional rotations A

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FLIGHT DYNAMICS, TOPIC 1 THREE DIMENSIONAL ROTATION AND REFERENCE FRAMES 1.1 The rotation and the direction cosine matrix A rotation is a rigid transformation that preserves the length, the volume (with sign), and the angle between pairs of vectors. If the vectors  is rotated into the vectors  , these properties are formally expressed by: •Conservation of the length:  ; •Conservation of the volume with sign:  ; •Conservation of the angle:  . It is remarkable that the composition of successive rotations can not be expressed by simply adding the corresponding rotation vectors and that rotations do not commute. In mechanics, rotations are used for expressing relationships between coordinate systems, called frames if the systems are rectangular. For instance, given two triads  and  , the latter can always be seen as obtained by a suitable rotation of the former. In order to transform the components between triads, the concept of direction cosine matrix is needed. The direction cosine matrix links the orthonormal triads of unit vectors  and  as follows:  Hence, the direction cosine matrix transforms one the generic unit vector  measured in  into the generic unit vector  measured in  and the generic unit vector  measured in  into the generic unit vector  measured in  . In order to obtain the expression of the direction cosine matrix, the generic unit vector  should be expressed as:  By generalizing, the generic component  is:  Hence, the direction cosine matrix can be written as follows: …