Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Appendix C / Axis Transforms
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APPENDIX C
16 AXIS TRANSFORMS
The orientation of forces and moments acting at a point are expressed in
different Frames of Reference in terms of direction cosines. The
transformation matrices defined here using either elemental Euler rotations,
or the four-parameter method of quaternions described in §22.9 to avoid the
singularity that occurs at pitch angles of ± 90°. Additional notation is
introduced to deal with the set of 12 possible Euler angle triplets, although
only the yaw-pitch-roll and roll-pitch-yaw variants are considered for the
current application. Small angle approximations, the Euler skew-symmetric
form and its memory mapping using the FORTRAN computer language are
presented.
The transformations between Cartesian, Spherical and UVR co-ordinates
with respect to a given frame are explored. The UVR co-ordinate system is
of particular interest since phased-array radar usually provide measurements
comprising 2 direction cosines (U and V) and range; rarely do they output
bearing, elevation and range directly.
Finally, the primary axis transforms used in this application are defined, i.e.
transforms that cannot be expressed as the product of two or more direction
cosine matrices. When defining algorithms any transform used should be
related to a combination of these primary cases. Of particular interest is the
relationship between direction cosines defined using trigonometric functions
and those described using Cartesian linear position and velocity parameters
with their associated singularities.
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