universidad tecnolÓgica de la mixteca anÁlisis ...jupiter.utm.mx/~tesis_dig/13838.pdf ·...
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UNIVERSIDAD TECNOLÓGICA DE LA MIXTECA
INSTITUTO DE FÍSICA Y MATEMÁTICAS
ANÁLISIS CINEMÁTICO DE UN MÓDULO ROBÓTICO
PARALELO CON TRES GRADOS DE LIBERTAD
TESIS
PARA OBTENER EL TÍTULO DE:
INGENIERO EN FÍSICA APLICADA
PRESENTA:
MARTÍNEZ LUNA ABEL EMANUEL
DIRECTOR DE TESIS:
DR. EDUARDO PIÑA GARZA
CO-DIRECTOR DE TESIS:
DR. IVÁN RENÉ CORRALES MENDOZA
Huajuapan de León, Oaxaca. Febrero de 2019
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VII
VIII
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❖
❖
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❖
❖
❖
❖
❖
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15
16
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(𝑥, 𝑦, 𝑧)
𝒂
𝑹
18
𝒙
𝑹𝑇𝑹 = 𝑬
𝑬
Φ 𝒏
𝒙 = 𝑹
𝑂𝐴⃗⃗⃗⃗ ⃗
𝒙
𝑂𝑋⃗⃗ ⃗⃗ ⃗
𝑂𝑂´⃗⃗ ⃗⃗ ⃗⃗ ⃗
𝒙
19
𝑂𝐴⃗⃗⃗⃗ ⃗ 𝑂𝑂’⃗⃗⃗⃗⃗⃗ ⃗ 𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗
𝒙 𝑂𝑋⃗⃗ ⃗⃗ ⃗ 𝑂𝑂’⃗⃗⃗⃗⃗⃗ ⃗ 𝑂’𝑋⃗⃗ ⃗⃗ ⃗⃗
𝑂𝑂´⃗⃗ ⃗⃗ ⃗⃗ ⃗
𝑂𝑂´⃗⃗ ⃗⃗ ⃗⃗ ⃗ 𝒏𝒏𝑻
𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗
𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗ 𝒏𝒏𝑻 𝒏𝒏𝑻
𝑂’𝑋⃗⃗ ⃗⃗ ⃗⃗ 𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗
𝑂’𝐵⃗⃗⃗⃗⃗⃗ ⃗
𝑂’𝑋⃗⃗ ⃗⃗ ⃗⃗ 𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗ 𝑂’𝐵⃗⃗⃗⃗⃗⃗ ⃗
𝑂’𝐵⃗⃗⃗⃗⃗⃗ ⃗ 𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗
𝑂’𝐵⃗⃗⃗⃗⃗⃗ ⃗ × 𝑂’𝐴⃗⃗ ⃗⃗ ⃗⃗ × 𝒏𝒏𝑻 ×
20
𝑂’𝑋⃗⃗ ⃗⃗ ⃗⃗ 𝒏𝒏𝑻 ×
𝒙 𝒏𝒏𝑻 𝒏𝒏𝑻 ×
𝑹 𝒏𝒏𝑻 𝒏𝒏𝑻 ×
𝑹𝑻 𝒏𝒏𝑻 𝒏𝒏𝑻 ×
𝑋𝑌𝑍
𝑋𝑌𝑍 𝜙 (0 ≤ 𝜙 < 2𝜋) 𝑍
𝑍𝑌 𝜃 (0 ≤ 𝜃 < 𝜋) 𝑋
21
𝑋𝑥2𝑥3
𝜓 (0 ≤ 𝜓 < 2𝜋) 𝑥3
𝑥1𝑥2𝑥3 𝜙, 𝜃 𝑦 𝜓
𝑥1𝑥2𝑥3 𝑋𝑌𝑍
𝑹 = 𝑹𝟑𝒛(𝜓)𝑹2𝑥(𝜃)𝑹𝟏𝒛(𝜙)
Φ = 𝜙 𝒏 = [0 0 1]𝑻
22
𝑹𝟏𝒛 = (𝑐𝑜𝑠𝜙 −𝑠𝑒𝑛𝜙 0𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜙 00 0 1
)
𝑹2𝑥 𝒏 = [1 0 0]𝑻
𝑂𝑋 Φ = 𝜃 𝑹𝑥
𝑹2𝑥 = (1 0 00 𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃0 𝑠𝑒𝑛𝜃 𝑐𝑜𝑠𝜃
)
𝒏 = [0 0 1]𝑻
𝜙 𝜓
𝑹𝟑𝒛 = (𝑐𝑜𝑠𝜓 −𝑠𝑒𝑛𝜓 0𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓 00 0 1
)
𝑹
𝑹 = (𝑐𝑜𝑠𝜓 −𝑠𝑒𝑛𝜓 0𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓 00 0 1
)(1 0 00 𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃0 𝑠𝑒𝑛𝜃 𝑐𝑜𝑠𝜃
)(𝑐𝑜𝑠𝜙 −𝑠𝑒𝑛𝜙 0𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜙 00 0 1
)
23
𝑹 = (
𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜃 −𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 + 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜙
𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 𝑐𝑜𝑠𝜃)
𝑹1 𝑂𝑍 𝜙 [1 0 0]𝑇
[cos𝜙 sin𝜙 0]𝑇
𝜃 𝑂𝑍 𝜃
𝜓 [cos𝜙 sin𝜙 0]𝑇
[cos𝜓 sin𝜓 0]𝑇
24
25
26
𝐶2
{𝑃𝑘}1𝑁 {𝜌𝑘, 𝜃𝑘}1
𝑁 𝛼 𝛽
𝑙𝑣 𝑙ℎ
𝑘 = 1,… ,𝑁
𝜃1 = 𝛼, 𝜃𝑁 =𝜋
2− 𝛽
𝜃𝑘 = 𝛼 +𝑘−1
𝑁−1[𝜋
2− (𝛼 + 𝛽)]
𝜌1 = √𝑙𝑣2 + 𝑎2 𝜌𝑁 = √𝑙ℎ
2 + 𝑏2
𝑁 𝑁 − 2 𝜌
𝐶2
𝑃1 90° − 𝛼 𝑂𝑃1
𝑃𝑁 90° − 𝛽 𝑂𝑃𝑁
𝑃1 𝑃𝑁
𝛾 tan 𝛾 =𝜌(𝜃)
𝜌′(𝜃)
𝜅
27
𝜅 =𝜌2+2(𝜌′)2−𝜌𝜌′′
(𝜌2+(𝜌′)2)32⁄
𝜌1
𝜌1′= tan(
𝜋
2− 𝛼)
𝜌𝑁
𝜌𝑁′= tan(
𝜋
2+ 𝛽)
𝜌12 + 2(𝜌1′)
2 − 𝜌1𝜌1′′ = 0
𝜌𝑁2 + 2(𝜌𝑁′)
2 − 𝜌𝑁𝜌𝑁′′ = 0
𝜌 = 𝜌(𝜃)
𝑎 = 89.5 𝑚𝑚 𝑏 = 75.5 𝑚𝑚 𝑙𝑣 = 𝑙ℎ = 12 𝑚𝑚 𝑁 = 20
𝑥𝑖 𝑦𝑖
𝑖 𝑥𝑖[𝑚𝑚] 𝑦𝑖[𝑚𝑚] 𝜌𝑖[𝑚𝑚] 𝜅[𝑚𝑚] 𝑖 𝑥𝑖[𝑚𝑚] 𝑦𝑖[𝑚𝑚] 𝜌𝑖[𝑚𝑚] 𝜅[𝑚𝑚]
28
�̂�𝒊 𝑖 = 1,2,3
𝛼𝑖 �̂�𝒊
𝛼𝑖
29
�̂�𝒊 𝛼𝑖
𝑥𝑦𝑧
�̂�𝟏 �̂�𝟐 𝑦 �̂�𝟑
�̂�𝒊, �̂�𝒊 �̂�𝒊
�̂�𝒊
30
�̂�𝟏 �̂�𝟐 𝑦 �̂�𝟑
�̂�𝟏 = [1 0 0 ]𝑇
�̂�𝟐 = [0 1 0 ]𝑇
�̂�𝟑 = [0 0 1 ]𝑇
𝑇
�̂�𝟏
�̂�𝟏 𝛼1 �̂�𝟑
�̂�𝟐 �̂�𝟑
�̂�𝟏 = [0 − 𝑠𝑖𝑛𝛼1 𝑐𝑜𝑠𝛼1]𝑇
�̂�𝟐
�̂�𝟐 𝛼2 �̂�𝟏 �̂�𝟏
�̂�𝟑.
�̂�𝟐 = [𝑐𝑜𝑠𝛼2 0 − 𝑠𝑖𝑛𝛼2]𝑇
�̂�𝟑
�̂�𝟑 𝛼3 �̂�𝟐 �̂�𝟏
31
�̂�𝟐
�̂�𝟑 = [ −𝑠𝑖𝑛𝛼3 𝑐𝑜𝑠𝛼3 0]𝑇
�̂�𝟏 �̂�𝟏 �̂�𝟏
�̂�𝟏
�̂�𝟏
�̂�𝟐 �̂�𝟐 �̂�𝟐
�̂�𝟐
�̂�𝟐
�̂�𝟑 �̂�𝟑 �̂�𝟑
�̂�𝟑
�̂�𝟑
�̂�𝟏 �̂�𝟑 �̂�𝟏
𝑧
�̂�𝟏∗ = [0 − 1 0]𝑻
�̂�𝟐
�̂�𝟏 �̂�𝟐 𝑦
�̂�𝟐∗ = [0 0 − 1]𝑻
32
�̂�𝟑 �̂�𝟐 �̂�𝟑
𝑥
�̂�𝟑∗ = [−1 0 0]𝑻
𝑧𝑦𝑥
𝑥 𝑦
𝑧 �̂�𝒊∗ �̂�𝒊
𝑹 = 𝑹𝟏𝒛(𝜙)𝑹2𝑦(𝜃)𝑹𝟑𝒙(𝜓)
𝑹𝟏𝒛(𝜙) 𝑹2𝑦(𝜃)
33
𝒏 = [0 1 0]𝑻 Φ = 𝜃
𝑹2𝑦(𝜃) = (𝑐𝑜𝑠𝜃 0 𝑠𝑒𝑛𝜃0 1 0
−𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃)
𝑹𝟑𝒙(𝜓) 𝜃
𝜓
𝑹𝟑𝒙(𝜓) = (1 0 00 𝑐𝑜𝑠𝜓 −𝑠𝑒𝑛𝜓0 𝑠𝑖𝑛𝜓 𝑐𝑜𝑠𝜓
)
𝑹 = (𝑐𝑜𝑠𝜙 −𝑠𝑒𝑛𝜙 0𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜙 00 0 1
)(𝑐𝑜𝑠𝜃 0 𝑠𝑒𝑛𝜃0 1 0
−𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃)(1 0 00 𝑐𝑜𝑠𝜓 −𝑠𝑒𝑛𝜓0 𝑠𝑖𝑛𝜓 𝑐𝑜𝑠𝜓
)
𝜙, 𝜃 𝜓
𝑹 = (
𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 + 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓−𝑠𝑒𝑛𝜃 𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓
)
34
�̂�𝒊
𝑖 = 1,2,3
�̂�𝟏 = 𝑹�̂�𝟏∗ = (
𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓−𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓
−𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓)
�̂�𝟐 = 𝑹�̂�𝟐∗ = (
−𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓
−𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓)
�̂�𝟑 = 𝑹�̂�𝟑∗ = (
−𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃−𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜃
)
�̂�𝒊 �̂�𝒊
�̂�𝒊𝑻�̂�𝒊 = 0, 𝑖 = 1,2,3.
(𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓)𝑠𝑒𝑛𝛼1 − 𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝛼1 = 0
(−𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓)𝑐𝑜𝑠𝛼2 + 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓𝑠𝑒𝑛𝛼2 = 0
𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝛼3 − 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼3 = 0
35
tan𝛼1 =𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓
𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓+𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓
tan𝛼2 =𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓+𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓
𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓
tan 𝛼3 = 𝑡𝑎𝑛𝜙
−�̂�𝟑 −�̂�𝟏 −�̂�𝟐
�̂�𝒊
�̂�𝑖𝑘 �̂�𝒊
𝑐𝑜𝑠𝜃𝑠𝑒𝑛(𝛼3 − 𝜙) = 0
𝑐𝑜𝑠𝜃 = 0
𝑠𝑒𝑛(𝛼3 − 𝜙) = 0
tan𝛼1 =�̂�13
�̂�12
tan𝛼2 =�̂�21
�̂�23
tan𝛼3 =�̂�32
�̂�31
36
𝜃
𝜃 =𝜋
2𝜃 = −
𝜋
2
𝜃 =𝜋
2
𝛼1
𝑐𝑜𝑠(𝜙 − 𝜓) = 0
𝜃 = −𝜋
2
𝑐𝑜𝑠(𝜙 + 𝜓) = 0
𝜃 =𝜋
2
𝑹 = (0 −𝑠𝑒𝑛(𝜙 − 𝜓) 𝑐𝑜𝑠(𝜙 − 𝜓)
0 𝑐𝑜𝑠(𝜙 − 𝜓) 𝑠𝑒𝑛(𝜙 − 𝜓)−1 0 0
)
𝑹𝟏 = (0 −1 00 0 1−1 0 0
)
𝑹𝟐 = (0 1 00 0 −1−1 0 0
)
37
𝜃 = −𝜋
2
𝑹 = (0 −𝑠𝑒𝑛(𝜙 + 𝜓) −𝑐𝑜𝑠(𝜙 + 𝜓)0 𝑐𝑜𝑠(𝜙 + 𝜓) −𝑠𝑒𝑛(𝜙 + 𝜓)1 0 0
)
𝑹𝟑 = (0 −1 00 0 −11 0 0
)
𝑹𝟒 = (0 1 00 0 11 0 0
)
𝒄𝟏 = 𝑹𝟏𝒄𝟏∗ = [
100] 𝒄𝟐 = 𝑹𝟏𝒄𝟐
∗ = [0−10] 𝒄𝟑 = 𝑹𝟏𝒄𝟑
∗ = [001]
𝒄𝟏 = 𝑹𝟐𝒄𝟏∗ = [
−100] 𝒄𝟐 = 𝑹𝟐𝒄𝟐
∗ = [010] 𝒄𝟑 = 𝑹𝟐𝒄𝟑
∗ = [001]
38
𝒄𝟏 = 𝑹𝟑𝒄𝟏∗ = [
100] 𝒄𝟐 = 𝑹𝟑𝒄𝟐
∗ = [010] 𝒄𝟑 = 𝑹𝟑𝒄𝟑
∗ = [00−1]
𝒄𝟏 = 𝑹𝟒𝒄𝟏∗ = [
−100] 𝒄𝟐 = 𝑹𝟒𝒄𝟐
∗ = [0−10] 𝒄𝟑 = 𝑹𝟒𝒄𝟑
∗ = [00−1]
𝑠𝑒𝑛(𝛼3 − 𝜙) = 0 𝜙
𝜙 = 𝛼3 𝜙 = 𝛼3 ± 𝜋
𝜙 = 𝛼3
(𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼1)𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1𝑐𝑜𝑠𝜓 = 0
−𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑠𝑒𝑛𝜓 + (𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛼2 − 𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝜃)𝑐𝑜𝑠𝜓 = 0
{𝑝11𝑠𝑒𝑛𝜓 + 𝑝12𝑐𝑜𝑠𝜓 = 0𝑝21𝑠𝑒𝑛𝜓 + 𝑝22𝑐𝑜𝑠𝜓 = 0
39
𝑝11 = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼1
𝑝21 = −𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2
𝑝12 = 𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1
𝑝22 = 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛼2 − 𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝜃
𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓
𝑝11𝑝22 − 𝑝21𝑝12 = 0
𝑐𝑜𝑠𝜃(𝐴𝑠𝑒𝑛𝜃 + 𝐵𝑐𝑜𝑠𝜃) = 0
𝐴 = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼2𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3
𝐵 = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3 − 𝑐𝑜𝑠𝛼1𝑠𝑒𝑛𝛼2
𝑐𝑜𝑠𝜃 = 0
𝐴𝑠𝑒𝑛𝜃 + 𝐵𝑐𝑜𝑠𝜃 = 0
𝜃 −𝜋, 𝜋
𝜃 = tan−1 (−𝐵
𝐴) + 𝑘𝜋 𝑘 = 0,1
𝜃
𝜓 −𝜋, 𝜋
40
𝜓 = tan−1 (−𝑝12
𝑝11) + 𝑘𝜋 𝑘 = 0,1
𝜓 = tan−1 (−𝑝22
𝑝21) + 𝑘𝜋 𝑘 = 0,1
�̂�𝒊
𝛼𝑖
𝑠𝑒𝑛𝜃
𝑐𝑜𝑠𝜃=−𝐵
𝐴 (67)
𝐴 = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼2𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3
𝐵 = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3 − 𝑐𝑜𝑠𝛼1𝑠𝑒𝑛𝛼2
𝛼𝑖
𝑠𝑒𝑛2𝜃 + 𝑐𝑜𝑠2𝜃 =
1
𝑠𝑒𝑛2𝜃 =𝐵2
𝐴2𝑐𝑜𝑠2𝜃 = (1 − 𝑐𝑜𝑠2𝜃)
(1 +𝐵2
𝐴2 ) 𝑐𝑜𝑠2𝜃 = 1
𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃 =𝐴
±√𝐴2 + 𝐵2 (68)
𝑠𝑒𝑛𝜃 =−𝐵
±√𝐴2 + 𝐵2 (69)
41
�̂�𝟑
�̂�𝟑 = (−𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃−𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜃
) =1
±√𝐴2 + 𝐵2(−𝐴𝑐𝑜𝑠𝜙−𝐴𝑠𝑒𝑛𝜙−𝐵
) =−1
±√𝐴2 + 𝐵2(𝐴𝑐𝑜𝑠𝛼3𝐴𝑠𝑒𝑛𝛼3𝐵
)
�̂�𝟑 =−1
±√𝐴2 + 𝐵2(𝐴𝑐𝑜𝑠𝛼3𝐴𝑠𝑒𝑛𝛼3𝐵
) (70)
𝜙 = 𝛼3
�̂�𝟑
𝑝11 𝑝22
𝑝11 = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼1 =−1
±√𝐴2 + 𝐵2(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1) (71)
𝑝22 = 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛼2 − 𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝜃 =1
±√𝐴2 + 𝐵2(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3) (72)
𝑠𝑒𝑛𝜓 =−𝑝12𝑝11
𝑐𝑜𝑠𝜓 (73)
𝑠𝑒𝑛2𝜓 + 𝑐𝑜𝑠2𝜓 = 1 𝑐𝑜𝑠𝜓
𝑐𝑜𝑠𝜓 =𝑝11
±√𝑝112 + 𝑝122 (74)
𝑠𝑒𝑛𝜓 =−𝑝12
±√𝑝112 + 𝑝122 (75)
�̂�𝟏
�̂�𝟏 = (
𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓−𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓
−𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓)
42
�̂�𝟏 =
(
𝑝11𝑠𝑒𝑛𝜙
±√𝑝112 + 𝑝122−
𝐵𝑝12𝑐𝑜𝑠𝜙
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122)
−𝐵𝑝12𝑠𝑒𝑛𝜙
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122)−
𝑝11𝑐𝑜𝑠𝜙
±√𝑝112 + 𝑝122
𝐴𝑝12
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122) )
�̂�𝟏 =1
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122)(
√𝐴2 + 𝐵2𝑝11𝑠𝑒𝑛𝜙 − 𝐵𝑝12𝑐𝑜𝑠𝜙
𝐵𝑝12𝑠𝑒𝑛𝜙 − √𝐴2 + 𝐵2𝑝11𝑐𝑜𝑠𝜙𝐴𝑝12
)
𝑝11 𝑝12 = 𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1
𝜙 = 𝛼3 𝛿 = ±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122)
�̂�𝟏 =1
𝛿(−(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)𝑠𝑒𝑛𝛼3 − 𝐵𝑐𝑜𝑠
2𝛼3𝑠𝑒𝑛𝛼1𝐵𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + (𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)𝑐𝑜𝑠𝛼3
𝐴𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1
) (76)
�̂�𝟏 𝛿
�̂�𝟐
�̂�𝟐 = (
−𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓
−𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓) =
(
𝐵𝑐𝑜𝑠𝜙𝑝11
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122)+
𝑝12𝑠𝑒𝑛𝜙
±√𝑝112 + 𝑝122
−𝑝12𝑐𝑜𝑠𝜙
±√𝑝112 + 𝑝12
2+
𝐵𝑠𝑒𝑛𝜙𝑝11
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝12
2)
−𝐴𝑝11
±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122) )
�̂�𝟐 =1
𝛿(
𝐵𝑐𝑜𝑠𝜙𝑝11 + √𝐴2 + 𝐵2𝑝12𝑠𝑒𝑛𝜙
−√𝐴2 + 𝐴2𝑝12𝑐𝑜𝑠𝜙 + 𝐵𝑠𝑒𝑛𝜙𝑝11−𝐴𝑝11
)
𝑝11 𝑝12 𝜙 = 𝛼3
43
�̂�𝟐 =1
𝛿
(
−𝐵𝑐𝑜𝑠𝛼3(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
±√𝐴2 + 𝐵2+√𝐴2 + 𝐵2𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3
−√𝐴2 + 𝐵2𝑠𝑒𝑛𝛼1𝑐𝑜𝑠2𝛼3 −
𝐵𝑠𝑒𝑛𝛼3(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
±√𝐴2 + 𝐵2
𝐴(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
±√𝐴2 + 𝐵2 )
1
±√𝐴2+𝐵2
�̂�𝟐 =1
𝛿√𝐴2+𝐵2(
−𝐵𝑐𝑜𝑠𝛼3(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1) + (𝐴2 + 𝐵2)𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3
−(𝐴2 + 𝐵2)𝑠𝑒𝑛𝛼1𝑐𝑜𝑠2𝛼3 − 𝐵𝑠𝑒𝑛𝛼3(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
𝐴(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
)
�̂�𝟏 =1
𝛿(−(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)𝑠𝑒𝑛𝛼3 − 𝐵𝑐𝑜𝑠
2𝛼3𝑠𝑒𝑛𝛼1𝐵𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + (𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)𝑐𝑜𝑠𝛼3
𝐴𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1
)
�̂�𝟐 =1
𝛿√𝐴2 + 𝐵2(
−𝐵𝑐𝑜𝑠𝛼3(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1) + (𝐴2 +𝐵2)𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3
−(𝐴2 + 𝐵2)𝑠𝑒𝑛𝛼1𝑐𝑜𝑠2𝛼3 − 𝐵𝑠𝑒𝑛𝛼3(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
𝐴(𝐵𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1)
)
�̂�𝟑 =−1
±√𝐴2 + 𝐵2(𝐴𝑐𝑜𝑠𝛼3𝐴𝑠𝑒𝑛𝛼3𝐵
)
44
𝑠𝑒𝑛𝜓 =−𝑝22𝑝21
𝑐𝑜𝑠𝜓 (79)
𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓
𝑝12 → 𝑝22 𝑝11 → 𝑝21
�̂�𝟏
�̂�𝟏 = (
𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓−𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓
−𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓)
�̂�𝟏 =
(
𝑝21𝑠𝑒𝑛𝜙
±√𝑝212 + 𝑝22
2 −
𝐵𝑝22𝑐𝑜𝑠𝜙
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝22
2)
−𝐵𝑝22𝑠𝑒𝑛𝜙
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222)−
𝑝21𝑐𝑜𝑠𝜙
±√𝑝212 + 𝑝222
𝐴𝑝22
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222) )
�̂�𝟏 =1
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222)(
−√𝐴2 + 𝐵2𝑝21𝑠𝑒𝑛𝜙 − 𝐵𝑝22𝑐𝑜𝑠𝜙
−𝐵𝑝22𝑠𝑒𝑛𝜙 + √𝐴2 + 𝐵2𝑝21𝑐𝑜𝑠𝜙𝐴𝑝22
)
𝑝21 = −𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2 𝑝22 𝜙 = 𝛼3
𝜌 = ±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222)
𝑐𝑜𝑠𝜓 =𝑝21
±√𝑝212 + 𝑝22
2 𝑠𝑒𝑛𝜓 =
−𝑝22
±√𝑝212 + 𝑝22
2
45
�̂�𝟏 =1
𝜌
(
−√𝐴2 + 𝐵2𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑠𝑒𝑛𝛼3 −
𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑐𝑜𝑠𝛼3
±√𝐴2 + 𝐵2
−𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑠𝑒𝑛𝛼3
±√𝐴2 + 𝐵2+√𝐴2 + 𝐵2𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3
𝐴(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)
±√𝐴2 + 𝐵2 )
1
±√𝐴2+𝐵2±√𝐴2 + 𝐵2 𝜌
�̂�𝟏 =1
𝜌√𝐴2+𝐵2(
−(𝐴2 + 𝐵2)𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑠𝑒𝑛𝛼3 − 𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑐𝑜𝑠𝛼3−𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑠𝑒𝑛𝛼3 + (𝐴
2 + 𝐵2)𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3𝐴(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)
)
�̂�𝟐
�̂�𝟐 = (
−𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓
−𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓) =
(
𝐵𝑐𝑜𝑠𝜙𝑝21
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222)+
𝑝22𝑠𝑒𝑛𝜙
±√𝑝212 + 𝑝222
−𝑝22𝑐𝑜𝑠𝜙
±√𝑝212 + 𝑝222+
𝐵𝑠𝑒𝑛𝜙𝑝21
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222)
−𝐴𝑝21
±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222) )
�̂�𝟐 =1
𝜌(
𝐵𝑐𝑜𝑠𝜙𝑝21 + √𝐴2 + 𝐵2𝑝22𝑠𝑒𝑛𝜙
−√𝐴2 + 𝐵2𝑝22𝑐𝑜𝑠𝜙 + 𝐵𝑠𝑒𝑛𝜙𝑝21−𝐴𝑝21
)
𝑝21 = −𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2 𝑝22 𝜙 = 𝛼3
�̂�𝟐 =1
𝜌(
−𝐵𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2 + (𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑠𝑒𝑛𝛼3−(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑐𝑜𝑠𝛼3 − 𝐵𝑠𝑒𝑛
2𝛼3𝑐𝑜𝑠𝛼2𝐴𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2
)
46
�̂�𝟏 =1
𝜌√𝐴2 + 𝐵2(
(𝐴2 + 𝐵2)𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑠𝑒𝑛𝛼3 + 𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑐𝑜𝑠𝛼3𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑠𝑒𝑛𝛼3 − (𝐴
2 + 𝐵2)𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3−𝐴(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)
)
�̂�𝟐 =1
𝜌(
−𝐵𝑐𝑜𝑠𝛼3𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2 + (𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑠𝑒𝑛𝛼3−(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3)𝑐𝑜𝑠𝛼3 − 𝐵𝑠𝑒𝑛
2𝛼3𝑐𝑜𝑠𝛼2𝐴𝑠𝑒𝑛𝛼3𝑐𝑜𝑠𝛼2
)
�̂�𝟑 =1
±√𝐴2 + 𝐵2(𝐴𝑐𝑜𝑠𝛼3𝐴𝑠𝑒𝑛𝛼3𝐵
)
47
α1 = α2 = α3 =π
2
48
49
𝐴 = 0
𝑓(𝛼1, 𝛼2, 𝛼3) = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼2𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3 = 0
𝑠𝑒𝑛𝛼2 = 0 𝑦 𝑐𝑜𝑠𝛼3 = 0
𝑠𝑒𝑛𝛼3 = 0 𝑦 𝑐𝑜𝑠𝛼1 = 0
𝑠𝑒𝑛𝛼1 = 0 𝑦 𝑐𝑜𝑠𝛼2 = 0
𝑝12 = 0 𝑝22 = 0 𝑠𝑒𝑛𝜓 = 0
𝛼3 =𝜋
2
𝛼3 = −𝜋
2
𝑹𝟏𝒂 = (0 −1 0𝑐𝑜𝑠𝜃 0 𝑠𝑒𝑛𝜃−𝑠𝑒𝑛𝜃 0 𝑐𝑜𝑠𝜃
)
𝑹𝟐𝒂 = (0 1 0
−𝑐𝑜𝑠𝜃 0 −𝑠𝑒𝑛𝜃−𝑠𝑒𝑛𝜃 0 𝑐𝑜𝑠𝜃
)
𝑝11 = 0
𝑝21 = 0 𝑐𝑜𝑠𝜓 = 0
𝑹𝟏𝒃 = (𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜃 00 0 −1
𝑠𝑒𝑛𝜃 𝑐𝑜𝑠𝜃 0)
50
𝑹𝟐𝒃 = (𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃 00 0 1
−𝑠𝑒𝑛𝜃 −𝑐𝑜𝑠𝜃 0)
𝑝11 = 0 𝑐𝑜𝑠𝜃 = 0
𝑹𝟏𝒄 = (0 −𝑠𝑒𝑛(𝜙 − 𝜓) 𝑐𝑜𝑠(𝜙 − 𝜓)0 𝑐𝑜𝑠(𝜙 − 𝜓) 𝑠𝑒𝑛(𝜙 − 𝜓)−1 0 0
)
𝑹𝟐𝒄 = (0 −𝑠𝑒𝑛(𝜙 − 𝜓) −𝑐𝑜𝑠(𝜙 − 𝜓)0 𝑐𝑜𝑠(𝜙 − 𝜓) −𝑠𝑒𝑛(𝜙 − 𝜓)1 0 0
)
𝑐𝑜𝑠𝜃 = 0
𝑐𝑜𝑠𝜃 = 0
51
52
53
54
55
.
56
57
.
58
𝑍𝑖−1𝑍0 𝑍𝑛−1