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## Coordinate Transformations

Because of the many coordinate systems used in the flight simulator, the simulator makes extensive use of coordinate transformations. Most coordinate systems in the simulator are Cartesian, so that the transformation is linear.

(A word on notation: this paper indicates which coordinate system a component belongs to by a superscript. For example, the velocity components of the airplane in body axes are uB,vB,wB. In local axes, the components are uL,vL,wL. The superscript is not used where there is no confusion about what coordinate system a quantity belongs to.)

To transform a velocity, or some other vector quantity, from one coordinate system to another, only a rotation is required. For example, the transformation of the wind velocity vector from local axes to body axes is simply:

 (7)

where .

Transforming points requires a rotation and translation. Transforming points between body and local axes has an additional complication as well. The body axes originate from the reference point of the airplane; however, because of dynamic considerations, the airplane's location is specified by the coordinates of its CG. This requires an additional translation from the CG to the reference point. The transformation of a point from body to local coordinates is:

 (8)

where is the point in question, and is the coordinates of the CG in the indicated coordinate system. The opposite transformation is:

 (9)

Next: Airplane Dynamics Up: Introduction Previous: Control Variables
Carl Banks
2000-08-11