ABSTRACT
S߰·ܵߠ ܥ߰·ܥ߶ െ ܵ߰·ܥߠ·ܵ߶ ܥ߰·ܵ߶ ܵ߰·ܥߠ·ܥ߶ C߰·ܵߠ െܵ߰·ܥ߶ െ ܥ߰·ܥߠ·ܵ߶ െܵ߰·ܵ߶ ܥ߰·ܥߠ·ܥ߶
൩
Sequence: 1-3-1
ܥߠ ܵߠ·ܥ߶ ܵߠ·ܵ߶
െC߰·ܵߠ C߰·ܥߠ·ܥ߶ െ ܵ߰·S߶ C߰·ܥߠ·ܵ߶ ܵ߰·C߶ S߰·ܵߠ െS߰·ܥߠ·ܥ߶ െ ܥ߰·S߶ െS߰·ܥߠ·ܵ߶ ܥ߰·C߶
൩
Sequence: 2-1-2
ܥ߰·ܥ߶ െ ܵ߰·ܥߠ·ܵ߶ S߰·ܵߠ െܥ߰·ܵ߶ െ ܵ߰·ܥߠ·ܥ߶
ܵߠ·ܵ߶ ܥߠ ܵߠ·ܥ߶ ܵ߰·ܥ߶ ܥ߰·ܥߠ·ܵ߶ െC߰·ܵߠ െܵ߰·ܵ߶ ܥ߰·ܥߠ·ܥ߶
൩
Sequence: 2-3-2 C߰·ܥߠ·ܥ߶ െ ܵ߰·S߶ C߰·ܵߠ െC߰·ܥߠ·ܵ߶ െ ܵ߰·C߶
െܵߠ·ܥ߶ ܥߠ ܵߠ·ܵ߶ S߰·ܥߠ·ܥ߶ ܥ߰·S߶ S߰·ܵߠ െS߰·ܥߠ·ܵ߶ ܥ߰·C߶
൩
Sequence: 3-1-3
ܥ߰·ܥ߶ െ ܵ߰·ܥߠ·ܵ߶ ܥ߰·ܵ߶ ܵ߰·ܥߠ·ܥ߶ S߰·ܵߠ
െܵ߰·ܥ߶ െ ܥ߰·ܥߠ·ܵ߶ െܵ߰·ܵ߶ ܥ߰·ܥߠ·ܥ߶ C߰·ܵߠ ܵߠ·ܵ߶ െܵߠ·ܥ߶ ܥߠ
൩
Sequence: 3-2-3
C߰·ܥߠ·ܥ߶ െ ܵ߰·S߶ C߰·ܥߠ·ܵ߶ ܵ߰·C߶ െC߰·ܵߠ
െS߰·ܥߠ·ܥ߶ െ ܥ߰·S߶ െS߰·ܥߠ·ܵ߶ ܥ߰·C߶ S߰·ܵߠ ܵߠ·ܥ߶ Sߠ·ܵ߶ ܥߠ
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Rotation Angle about Fixed-Axis Vector (see p. 338 for ሾࡹሿ)
Φ ൌ ܿݏିଵ ቆ12 ሺܯଵଵ ܯଶଶ ܯଷଷ െ 1ሻቇ
Fixed-Axis Vector (see p. 338 for ሾࡹሿ)
ࢋሬԦ ൌ ݁௫݁௬݁௭
൩ ൌ 12 · sin ሺΦሻ · ൭ ܯଶଷ െ ܯଷଶܯଷଵ െ ܯଵଷܯଵଶ െ ܯଶଵ
൱
Euler Symmetric Parameters ݍଵ ൌ ݁௫ · ݏ݅݊ ൬Φ2 ൰
ݍଶ ൌ ݁௬ · ݏ݅݊ ൬Φ2 ൰
ݍଷ ൌ ݁௭ · ݏ݅݊ ൬Φ2 ൰
ݍସ ൌ ܿݏ ൬Φ2 ൰ ݍଵଶݍଶଶݍଷଶݍସଶ ൌ 1
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ሾࡹሺሬԦሻሿ ൌ ܯଵଵ ܯଵଶ ܯଵଷ ܯଶଵ ܯଶଶ ܯଶଷܯଷଵ ܯଷଶ ܯଷଷ
൩
ܯଵଵ ൌ ݍଵଶെݍଶଶെݍଷଶݍସଶ ܯଶଷ ൌ 2 · ሺݍଶ · ݍଷ ݍଵ · ݍସሻ
ܯଵଶ ൌ 2 · ሺݍଵ · ݍଶ ݍଷ · ݍସሻ ܯଷଵ ൌ 2 · ሺݍଵ · ݍଷ ݍଶ · ݍସሻ
ܯଵଷ ൌ 2 · ሺݍଵ · ݍଷ െ ݍଶ · ݍସሻ ܯଷଶ ൌ 2 · ሺݍଶ · ݍଷ െ ݍଵ · ݍସሻ
ܯଶଵ ൌ 2 · ሺݍଵ · ݍଶ െ ݍଷ · ݍସሻ ܯଷଷ ൌ െݍଵଶെݍଶଶݍଷଶݍସଶ
Euler Symmetric Parameters Corresponding to a Given Rotation Matrix ሾࡹሿ ݍଵ ൌ 14 · ݍସ · ሺܯଶଷ െ ܯଷଶሻ
ݍଶ ൌ 14 · ݍସ · ሺܯଷଵ െ ܯଵଷሻ
ݍଷ ൌ 14 · ݍସ · ሺܯଵଶ െ ܯଶଵሻ
ݍସ ൌ േ 12 · ඥ1 ܯଵଵ ܯଶଶ ܯଷଷ
Note: Sign Ambiguity. Changing sign of all Euler Symmetric Parameters simultaneously does not affect the rotation matrix. As such, there are four possible ways to compute the Euler Symmetric Parameters.
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Rotation Angle about Fixed-Axis Vector (see p. 340 for ሾࡹሿ)
Φ ൌ ܿݏିଵ ቆ12 ሺܯଵଵ ܯଶଶ ܯଷଷ െ 1ሻቇ
Fixed-Axis Vector (see p. 340 for ሾࡹሿ)
ࢋሬԦ ൌ ݁௫݁௬݁௭
൩ ൌ 12 · sin ሺΦሻ · ൭ ܯଶଷ െ ܯଷଶܯଷଵ െ ܯଵଷܯଵଶ െ ܯଶଵ
൱
Gibbs Vector ݃ଵ ൌ ݍଵݍସ ൌ ݁௫ · ݐܽ݊ ൬
Φ 2 ൰
݃ଶ ൌ ݍଶݍସ ൌ ݁௬ · ݐܽ݊ ൬ Φ 2 ൰
݃ଷ ൌ ݍଷݍସ ൌ ݁௭ · ݐܽ݊ ൬ Φ 2 ൰
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ሾࡹሺࢍሬሬԦሻሿ ൌ ܯଵଵ ܯଵଶ ܯଵଷ ܯଶଵ ܯଶଶ ܯଶଷܯଷଵ ܯଷଶ ܯଷଷ
൩ ܯଵଵ ൌ 1 ݃ଵ
2 · ሺ݃ଶ · ݃ଷ ݃ଵሻ 1 ݃ଵଶ݃ଶଶ݃ଷଶ
ܯଵଶ ൌ 2 · ሺ݃ଵ · ݃ଶ ݃ଷሻ
1 ݃ଵଶ݃ଶଶ݃ଷଶ ܯଷଵ ൌ 2 · ሺ݃ଵ · ݃ଷ ݃ଶሻ 1 ݃ଵଶ݃ଶଶ݃ଷଶ
ܯଵଷ ൌ 2 · ሺ݃ଵ · ݃ଷ െ ݃ଶሻ
1 ݃ଵଶ݃ଶଶ݃ଷଶ ܯଷଶ ൌ 2 · ሺ݃ଶ · ݃ଷ െ ݃ଵሻ 1 ݃ଵଶ݃ଶଶ݃ଷଶ
ܯଶଵ ൌ 2 · ሺ݃ଵ · ݃ଶ െ ݃ଷሻ
1 ݃ଵଶ݃ଶଶ݃ଷଶ
Gibbs Vector Corresponding to a Given Rotation Matrix ሾࡹሿ ݃ଵ ൌ ܯଶଷ െ ܯଷଶ1 ܯଵଵ ܯଶଶ ܯଷଷ
݃ଶ ൌ ܯଷଵ െ ܯଵଷ1 ܯଵଵ ܯଶଶ ܯଷଷ
݃ଷ ൌ ܯଵଶ െ ܯଶଵ1 ܯଵଵ ܯଶଶ ܯଷଷ ______________________________________________________________________
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