ABSTRACT
Sequence: 1-2-3 ߶ሶ ൌ ቀ߱௫ · ܿݏሺ߰ሻ െ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ݏ݁ܿሺߠሻ ߠሶ ൌ ߱௬ · ܿݏሺ߰ሻ ߱௫ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௭ െ ቀ߱௫ · ܿݏሺ߰ሻ െ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ݐܽ݊ሺߠሻ
Sequence: 1-3-2 ߶ሶ ൌ ൫߱௫ · ܿݏሺ߰ሻ ߱௭ · ݏ݅݊ሺ߰ሻ൯ · ݏ݁ܿሺߠሻ ߠሶ ൌ ߱௭ · ܿݏሺ߰ሻ െ ߱௫ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௬ ൫߱௫ · ܿݏሺ߰ሻ ߱௭ · ݏ݅݊ሺ߰ሻ൯ · ݐܽ݊ሺߠሻ
Sequence: 2-3-1 ߶ሶ ൌ ቀ߱௬ · ܿݏሺ߰ሻ െ ߱௭ · ݏ݅݊ሺ߰ሻቁ · ݏ݁ܿሺߠሻ ߠሶ ൌ ߱௭ · ܿݏሺ߰ሻ ߱௬ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௫ െ ቀ߱௬ · ܿݏሺ߰ሻ െ ߱௭ · ݏ݅݊ሺ߰ሻቁ · ݐܽ݊ሺߠሻ
Sequence: 2-1-3 ߶ሶ ൌ ቀ߱௬ · ܿݏሺ߰ሻ ߱௫ · ݏ݅݊ሺ߰ሻቁ · ݏ݁ܿሺߠሻ ߠሶ ൌ ߱௫ · ܿݏሺ߰ሻ െ ߱௬ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௭ ቀ߱௬ · ܿݏሺ߰ሻ ߱௫ · ݏ݅݊ሺ߰ሻቁ · ݐܽ݊ሺߠሻ
Sequence: 3-1-2 ߶ሶ ൌ ൫߱௭ · ܿݏሺ߰ሻ െ ߱௫ · ݏ݅݊ሺ߰ሻ൯ · ݏ݁ܿሺߠሻ ߠሶ ൌ ߱௫ · ܿݏሺ߰ሻ ߱௭ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௬ െ ൫߱௭ · ܿݏሺ߰ሻ െ ߱௫ · ݏ݅݊ሺ߰ሻ൯ · ݐܽ݊ሺߠሻ
Sequence: 3-2-1 ߶ሶ ൌ ቀ߱௭ · ܿݏሺ߰ሻ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ݏ݁ܿሺߠሻ ߠሶ ൌ ߱௬ · ܿݏሺ߰ሻ െ ߱௭ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௫ ቀ߱௭ · ܿݏሺ߰ሻ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ݐܽ݊ሺߠሻ
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Sequence: 1-2-1 ߶ሶ ൌ ቀ߱௭ · ܿݏሺ߰ሻ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ܿݏܿሺߠሻ ߠሶ ൌ ߱௬ · ܿݏሺ߰ሻ െ ߱௭ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௫ െ ቀ߱௭ · ܿݏሺ߰ሻ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ܿݐሺߠሻ
Sequence: 1-3-1 ߶ሶ ൌ െ ቀ߱௬ · ܿݏሺ߰ሻ െ ߱௭ · ݏ݅݊ሺ߰ሻቁ · ܿݏܿሺߠሻ ߠሶ ൌ ߱௭ · ܿݏሺ߰ሻ ߱௬ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௫ ቀ߱௬ · ܿݏሺ߰ሻ െ ߱௭ · ݏ݅݊ሺ߰ሻቁ · ܿݐሺߠሻ
Sequence: 2-1-2 ߶ሶ ൌ െ൫߱௭ · ܿݏሺ߰ሻ െ ߱௫ · ݏ݅݊ሺ߰ሻ൯ · ܿݏܿሺߠሻ ߠሶ ൌ ߱௫ · ܿݏሺ߰ሻ ߱௭ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௬ ൫߱௭ · ܿݏሺ߰ሻ െ ߱௫ · ݏ݅݊ሺ߰ሻ൯ · ܿݐሺߠሻ
Sequence: 2-3-2 ߶ሶ ൌ ൫߱௫ · ܿݏሺ߰ሻ ߱௭ · ݏ݅݊ሺ߰ሻ൯ · ܿݏܿሺߠሻ ߠሶ ൌ ߱௭ · ܿݏሺ߰ሻ െ ߱௫ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௬ െ ൫߱௫ · ܿݏሺ߰ሻ ߱௭ · ݏ݅݊ሺ߰ሻ൯ · ܿݐሺߠሻ
Sequence: 3-1-3 ߶ሶ ൌ ቀ߱௬ · ܿݏሺ߰ሻ ߱௫ · ݏ݅݊ሺ߰ሻቁ · ܿݏܿሺߠሻ ߠሶ ൌ ߱௫ · ܿݏሺ߰ሻ െ ߱௬ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௭ െ ቀ߱௬ · ܿݏሺ߰ሻ ߱௫ · ݏ݅݊ሺ߰ሻቁ · ܿݐሺߠሻ
Sequence: 3-2-3 ߶ሶ ൌ െ ቀ߱௫ · ܿݏሺ߰ሻ െ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ܿݏܿሺߠሻ ߠሶ ൌ ߱௬ · ܿݏሺ߰ሻ ߱௫ · ݏ݅݊ሺ߰ሻ
ሶ߰ ൌ ߱௭ ቀ߱௫ · ܿݏሺ߰ሻ െ ߱௬ · ݏ݅݊ሺ߰ሻቁ · ܿݐሺߠሻ
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Skew Symmetric Matrix
Ω ൌ ൦ 0 ߱௭ െ߱௬ ߱௫
െ߱௭ 0 ߱௫ ߱௬߱௬െ߱௫ െ߱௫െ߱௬
0 െ߱௭
߱௭0 ൪
Quaternion Kinematic Equation of Motion (see p. 337 for ሬԦ) ݀ሬԦ ݀ݐ ൌ
1 2 · Ω · ሬԦ ൌ
1 2 · Ω ·
ݍଵݍଶݍଷݍସ
Gibbs Vector Kinematic Equation of Motion (see p. 339 for ࢍሬሬԦ) ݀ࢍሬሬԦ ݀ݐ ൌ
1 2 · ሺ࣓ሬሬሬԦ െ ࣓ሬሬሬԦ ൈ ࢍሬሬԦ ሺ࣓ሬሬሬԦ · ࢍሬሬԦሻ · ࢍሬሬԦሻ
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Angular Momentum (see pp. 349 and 350 for ࣓ሬሬሬԦ) ࡴሬሬሬԦ ൌ ܫ௫ · ߱௫ଙ̂ ܫ௬ · ߱௬ଚ̂ ܫ௭ · ߱௭ หࡴሬሬሬԦห ൌ ܫ௭ · ߱௭ܿݏሺߠሻ ൌ ܫ௫ · ห࣓ሬሬሬԦห
Units:
ࡴሬሬሬԦ ൌ ሺ݉ܽݏݏሻ · ሺ݈݁݊݃ݐ݄ଶሻ ሺݐ݅݉݁ሻ⁄ ܫ௫ ൌ ܫ௬ ൌ ܫ௭ ൌ ሺ݉ܽݏݏሻ · ሺ݈݁݊݃ݐ݄ଶሻ
߱௫ ൌ ߱௬ ൌ ߱௭ ൌ ࣓ሬሬሬԦ ൌ ࣓ሬሬሬԦ ൌ ࣓ሬሬሬԦ࢙ ൌ Radian ሺݐ݅݉݁ሻ⁄
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ห࣓ሬሬሬԦห ൌ ܫ௭ܫ௫ െ ܫ௭ ·
|࣓ሬሬሬԦ࢙| ܿݏሺߠሻ
|࣓ሬሬሬԦ࢙| ൌ ܫ௫ െ ܫ௭ܫ௫ · ߱௭
ܿݏሺߛሻ ൌ ߱௭|࣓ሬሬሬԦ| ൌ
ܿݏሺߠሻ
ܫ௫ଶ ൬1 െ ܫ௭ଶ ܫ௫ଶ൰ · ܿݏ
ଶሺߠሻ
ࡴሬሬሬԦ
࣓ሬሬሬԦ
࣓ሬሬሬԦ࢙ ࣓ሬሬሬԦ
Space Cone
Body Cone
C.G. ܫ௫ ൌ ܫ௬ ܫ௭
ݖ
ߠ ߛ
Moments of Inertia
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ห࣓ሬሬሬԦห ൌ ܫ௭ܫ௫ െ ܫ௭ ·
|࣓ሬሬሬԦ࢙| ܿݏሺߠሻ
|࣓ሬሬሬԦ࢙| ൌ ܫ௫ െ ܫ௭ܫ௫ · ߱௭
ܿݏሺߛሻ ൌ ߱௭|࣓ሬሬሬԦ| ൌ
ܿݏሺߠሻ
ܫ௫ଶ ൬1 െ ܫ௭ଶ ܫ௫ଶ൰ · ܿݏ
ଶሺߠሻ
ࡴሬሬሬԦ
࣓ሬሬሬԦ ࣓ሬሬሬԦ࢙ Space Cone
Body Cone
C.G
ܫ௫ ൌ ܫ௬ ൏ ܫ௭
ݖ
࣓ሬሬሬԦ
ߛ
Moments of Inertia
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