UNIVERSIDAD TECNOLÓGICA DE LA MIXTECA ANÁLISIS ...jupiter.utm.mx/~tesis_dig/13838.pdf ·...

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

12

❖

❖

13

❖

❖

❖

❖

❖

17

(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

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


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


) (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 = (



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 = (


−𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓) =

(

𝐵𝑐𝑜𝑠𝜙𝑝11

±√(𝐴2 + 𝐵2)(𝑝112 + 𝑝122)+


±√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)


)

�̂�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)


)

�̂�𝟑 =−1


)

44

𝑠𝑒𝑛𝜓 =−𝑝22𝑝21

𝑐𝑜𝑠𝜓 (79)

𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓

𝑝12 → 𝑝22 𝑝11 → 𝑝21

�̂�𝟏

�̂�𝟏 = (



�̂�𝟏 =

(


±√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 = (


−𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓) =

(

𝐵𝑐𝑜𝑠𝜙𝑝21

±√(𝐴2 + 𝐵2)(𝑝212 + 𝑝222)+


±√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


)

47

α1 = α2 = α3 =π

2

49

𝐴 = 0

𝑓(𝛼1, 𝛼2, 𝛼3) = 𝑠𝑒𝑛𝛼1𝑠𝑒𝑛𝛼2𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1𝑐𝑜𝑠𝛼2𝑐𝑜𝑠𝛼3 = 0

𝑠𝑒𝑛𝛼2 = 0 𝑦 𝑐𝑜𝑠𝛼3 = 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

)



58

𝑍𝑖−1𝑍0 𝑍𝑛−1

UNIVERSIDAD TECNOLÓGICA DE LA MIXTECA ANÁLISIS ...jupiter.utm.mx/~tesis_dig/13838.pdf ·...

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Transcript of UNIVERSIDAD TECNOLÓGICA DE LA MIXTECA ANÁLISIS ...jupiter.utm.mx/~tesis_dig/13838.pdf ·...