tutorial1 - Rotations and Translations Representing a Point...

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Rotations and Translations
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Representing a Point 3D A point in space defined by vector P in frame A is given by A is a reference coordinate system here z y x A p p p P
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Representing a Point 3D (cont.) Once a coordinate system is fixed, we can locate any point in the universe with a 3x1 position vector. The components of P in { A } have numerical values which indicate distances along the axes of { A }. To describe the orientation of a body we will attach a coordinate system to the body and then give a description of this coordinate system relative to the reference system .
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Description of Orientation ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A B A B A B A B A B A B A B A B A B B A B A B A A B Z Z Z Y Z X Y Z Y Y Y X X Z X Y X X Z Y X R B X ˆ A X ˆ B X ˆ A Y ˆ B Y ˆ is a unit vector in B is a coordinate of a unit vector of B in coordinates system A (i.e. the projection of onto the unit direction of its reference) B A X ˆ B X ˆ
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Example Rotating B relative to A around Z by 30 000 . 1 000 . 0 000 . 0 000 . 0 866 . 0 500 . 0 000 . 0 500 . 0 866 . 0 R A B A X ˆ B X ˆ A Y ˆ B Y ˆ
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Example In general: A X ˆ B X ˆ A Y ˆ B Y ˆ 1 0 0 0 cos sin 0 sin cos Z A B R
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Using Rotation Matrices P R P B A B A
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Example: Pure rotation 0 2 1 P B ? P A 000 . 0 232 . 2 134 . 0 0 2 1 * 000 . 1 000 . 0 000 . 0 000 . 0 866 . 0 500 . 0 000 . 0 500 . 0 866 . 0 P A A X ˆ B X ˆ A Y ˆ B Y ˆ P B 30 P R P B A B A
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Translation A X ˆ A Y ˆ A Z ˆ B X ˆ B Y ˆ B Z ˆ P B ORG B A P ORG B A B A P P P
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Combining Rotation and Translation ORG B A B A B A P P R P A X ˆ B X ˆ P B ORG B A P P A A Y ˆ B Y ˆ
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This note was uploaded on 05/12/2011 for the course ME 427 taught by Professor Vedattemiz during the Spring '11 term at Işık Üniversitesi.

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tutorial1 - Rotations and Translations Representing a Point...

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