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Definition of the Euler Transformation Matrices

To transform motions about the Mir body axes into the inertial reference frame, define three Euler angles [Graphics:../Images/index_gr_1.gif], [Graphics:../Images/index_gr_2.gif], and [Graphics:../Images/index_gr_3.gif] as arguments of three rotation matrices. These matrices will be combined into one matrix to provide for a general multiaxis rotation. The Euler matrices [Graphics:../Images/index_gr_4.gif] [Graphics:../Images/index_gr_5.gif], and [Graphics:../Images/index_gr_6.gif] rotate from the inertial unprimed frame [Graphics:../Images/index_gr_7.gif] to the primed frame [Graphics:../Images/index_gr_8.gif] use the form [Graphics:../Images/index_gr_9.gif] where [Graphics:../Images/index_gr_10.gif] is a vector in inertial space. First rotate about the inertial space axis z to get to the first primed frame.

[Graphics:../Images/index_gr_11.gif]

At this point [Graphics:../Images/index_gr_12.gif]. Now we rotate about the primed frame y axis, by [Graphics:../Images/index_gr_13.gif].

[Graphics:../Images/index_gr_14.gif]

At this point  [Graphics:../Images/index_gr_15.gif]. Now rotate the double primed frame axis about the primed frame z axis (note, not x, since this is the Euler system).

[Graphics:../Images/index_gr_16.gif]

Finally [Graphics:../Images/index_gr_17.gif] . Define [Graphics:../Images/index_gr_18.gif] to be the combined solid body rotation matrix [Graphics:../Images/index_gr_19.gif] and [Graphics:../Images/index_gr_20.gif] be its inverse such that [Graphics:../Images/index_gr_21.gif]

[Graphics:../Images/index_gr_22.gif]
[Graphics:../Images/index_gr_23.gif]


Converted by Mathematica      October 6, 1999

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