Elem_rotations

Elem_rotations - f 2 = ˆ e 2(6 ˆ f 3 = sin θ ˆ e 1 cos...

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MAE146: Astronautics – Notes Benjamin F. Villac January 8, 2010 Given a reference frame R = (0 , ˆ e 1 , ˆ e 2 , ˆ e 3 ), the elementary rotations are deFned as the rotations transforming the basis ( ˆ e 1 , ˆ e 2 , ˆ e 3 ) into a basis ( ˆ f 1 , ˆ f 2 , ˆ f 3 ) with one of the basis vector Fxed (that is ˆ f i = ˆ e i for some index i ). The other basis vectors are rotated by a Fxed angle, θ , with positive angle being understood as counter-clock-wise rotations when looking down the Fxed vector. Rotation about the 1 st axis: The rotation leaving the 3 rd axis Fxed is deFned by the relations: ˆ f 1 = ˆ e 1 (1) ˆ f 2 = cos( θ ) . ˆ e 2 + sin( θ ) . ˆ e 3 (2) ˆ f 3 = - sin( θ ) . ˆ e 2 + cos( θ ) . ˆ e 3 (3) or, expressed in matrix form: ˆ f 1 ˆ f 2 ˆ f 3 = R 1 ( θ ) . ˆ e 1 ˆ e 2 ˆ e 3 where R 1 ( θ ) = 1 0 0 0 cos θ sin θ 0 - sin θ cos θ (4) Rotation about the 2 nd axis: The rotation leaving the 2 rd axis Fxed is deFned by the relations: ˆ f 1 = cos( θ ) . ˆ e 1 + - sin( θ ) . ˆ e 3 (5) ˆ
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Unformatted text preview: f 2 = ˆ e 2 (6) ˆ f 3 = sin( θ ) . ˆ e 1 + cos( θ ) . ˆ e 3 (7) or, expressed in matrix form: ˆ f 1 ˆ f 2 ˆ f 3 = R 2 ( θ ) . ˆ e 1 ˆ e 2 ˆ e 3 where R 2 ( θ ) = cos θ-sin θ 1 sin θ cos θ (8) Rotation about the 3 nd axis: The rotation leaving the 3 rd axis Fxed is deFned by the relations: ˆ f 1 = cos( θ ) . ˆ e 1 + sin( θ ) . ˆ e 2 (9) ˆ f 2 =-sin( θ ) . ˆ e 1 + cos( θ ) . ˆ e 2 (10) ˆ f 3 = ˆ e 3 (11) or, expressed in matrix form: ˆ f 1 ˆ f 2 ˆ f 3 = R 3 ( θ ) . ˆ e 1 ˆ e 2 ˆ e 3 where R 3 ( θ ) = cos θ sin θ-sin θ cos θ 1 (12) 1...
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This note was uploaded on 03/13/2010 for the course MAE 19100 taught by Professor Villac during the Winter '10 term at UC Irvine.

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