DISENO Y CONSTRUCCION DE UN MANIPULADOR TIPO SCARA CON TRES GRADOS DE LIBERTAD

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ESCUELA SUPERIOR DE INGENIERÍA

MECÁNICA Y ELÉCTRICA

“DISEÑO Y CONSTRUCCIÓN DE UN MANIPULADOR TIPO SCARA

CON TRES GRADOS DE LIBERTAD”

TESIS

QUE PARA OBTENER EL TÍTULO DE

INGENIERO EN CONTROL Y AUTOMATIZACIÓN

PRESENTAN

ISMAEL BASILIO VÁZQUEZ

GERARDO REYES SÁNCHEZ DEL VALLE

ASESORES

M. en C. GUSTAVO PACHECO VAN DYCK

M. en C. JUAN JOSÉ MUÑOZ CESAR

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La robótica es un campo relativamente nuevo de tecnología moderna, eso cruza los límites de la ingeniería tradicional. Entendiendo la complejidad de robots y sus aplicaciones requieren conocimiento de ingeniería eléctrica, mecánica, industrial, ciencias de la computación, economía, y matemáticas. Nuevas disciplinas de la ingeniería, como, la ingeniería de fabricación, aplicaciones de ingeniería, y la ingeniería de la ciencia, está empezando a surgir para tratar con la complejidad del campo de la robótica y el área más grande de la fábrica de la automatización. Dentro de unos pocos años es posible que el diseño de la ingeniería robótica estará sobre su propio pie, como una disciplina de ingeniería distinta_.

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(18)

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(19)

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(21)

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(22)

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(23)

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(24)

#

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,@*B02,/0*4@,4.*3,/-++080+0,3,,+12*3,274/041+-/032,@=-/03,240@'12, 041+003=+,E-63,@,4-,,+12*3,274/03,@*B02,052,550@.+0+-1,1-+,52,/,0E@'5 2-@H4 04 03 /50B- /0 3-5 @,4.*3,/-+05 2-4 @*B02,5 058;+2,5I .-+ >*0 4-5-1+-5 /04-1,@-5 @*B02,2*6-50A05/03,,+12*3,274410+20.1,4,.*41-52-@*40534244,13,2+-4₆₀₃ @,4.*3,/-+1,48-+/,@=-5104041+05(+,/-5/03=0+1,//0@*B02,058;+2,,@*B02,058;+2, [email protected], 03 ,4'355 /0 3, 240@'12, (+,4/0@0410I 082,E@0410 .0+@104/- ;3 /05,2-.30 /03 .-52-4,@041-63,-+041,274/0*4-=A01-<.2,@0410.-+2-45(*0410I03@,4.*3,/-+.-500 1+05.-52-405/0(+,/-5/03=0+1,/032*,305.+-/*2/-.-+1+05-@'5,+12*3,2-4050403=+,E-34H@0+-/0-+041,274/03-5(+,/-5/03=0+1,//0.04/0/03-5(+,/-5/03=0+1,//03,@*B02, 52-@H4042-41+,+@*B02,52-4*4,I/-5-1+05(+,/-5/03=0+1,//0.04/04/-/03,,.32,274

0D,/2D->*0*4+-=-1055-3-=*04-2-@-@,4--2-@-030@041-84,30*5,03=+,E-603045,@=30/03,@*B02,/0*4+-=-1.+@0+,@0410.,+,.-5274/03030@041-84,362*,3>*0+ D0++,@041,>*0.*0/,330,+503030@041-84,3-D0++,@041,3,>*0+0,3@0410+0,3E,031+,=,A--51.-5/05510@,5/05*A0274/0*4+-=-150@*051+,4043,5(*04101,=3,

3/.;,67<,2345 223.5193,58. 83-3=12345

4E,/0.+0574J

• 05.3,4(*3,+

• 05.3340,3

• 0*@'12-•

+,45.-+106@,4.*3,274/0.0E,5 5-=+03,5>*[email protected]+10.+05-4,+

4E,/004(,42D0 • 0*@'12-•

3;21+2- 0E,5/0(+,4/05/@045-405-5-=+03,5>*04-50.*0/00A0+20+

.+0574

F041-5,/0,2<- • 0*@'12-

*0+.-52-45*.0+82035,.-2-.-+-5,92+51,3I.3'512-I;12:

3021+-@'4 • 3;21+2- 0E,580++-@,(4;12,5

-4502*0410@0410I2-45/0+,@-5*4+-=-1/0/-5053,=-4052-@-50@*051+,043,8(*+, $0(,/-040384,3/0@,4.*3,/-+05*4,D0++,@041,/0,3(*4,23,50I2-@-*4,@-30/-+,-*4,+*0/,2-+1,/-+,*.-40@-5/050,+4-5-1+-5@-0+03@,4.*3,/-+/05/0*4,.-5274D,2, *4, .-5274 /05/0 2*6- .*41- /03 +-=-1 05 03 5(*0410 2-41-+4- /0 3, 5*.0+820 , 6 *4, 03-2/,/2-451,410032*,3@,41040*4,8*0+E,.+0052+1,)4-+@,3,3,5*.0+820

(25)

)(*+,-$-=-1/0/-5053,=-405 %& 40@'12,/03+-=-1J51*/-/05*@-@041-2-4+05.021-,*45510@,/0+080+042,J - 052+.274 ,4,3<12, /03 @-@041- 05.,2,3 /03 +-=-1 2-@- *4, 8*4274 /03

[email protected] 03,2-405041+03,.-52746-+041,274/030?1+0@-/03+-=-193-2,3E,274:63-5 ,3-+05/05*52--+/04,/,5,+12*3,+05

• +-=30@,240@'12-/+021-J010+@4,+3,.-52746-+041,274/030?1+0@-/03+-=-1I2-4

+05.021- , *4 5510@, /0 2--+/04,/,5 /0 +080+042,I 2-4-2/-5 3-5 ,3-+05 /0 3,5 ,+12*3,2-40563-5.,+'@01+-5(0-@;1+2-5/03-5030@041-5/03+-=-1

• +-=30@, 240@'12- 40+5-J 010+@4,+ 3, 2-48(*+,274 >*0 /0=0 ,/-.1,+ 03 +-=-1 .,+, ,32,4E,+*4,.-52746-+041,2742-4-2/,5 • -/03-/80+042,39@,1+EQ,2-=,4,:J03,2-405041+03,503-2/,/05/03@-@041-/0 3,5,+12*3,2-40563,5/030?1+0@-/03+-=-1 ,-=104274/03@-/03-/03,240@'12,/+021,50.*0/0-=1040+/0/80+041058-+@,53,5 2*,30550@042-4,+,4,2-414*,274J )(*+,-%03,274>*00?510041+03,240@'12,/+021,63,240@'12,40+5, 1 1 2 2 ( , , , , , ) ( , , , , , ) . . . ( , , , , , ) n n

q f x y z q f x y z

q f x y z

α β γ α β γ α β γ = = = 1 2 1 2 1 2 1 2 1 2 1 2 ( , ,...., ) ( , ,...., ) ( , ,...., ) ( , ,...., ) ( , ,...., ) ( , ,...., ) x n y n z n n n n

x f q q q

y f q q q

z f q q q

f q q q

f q q q

f q q q

(26)

$ 3 .+@0+- , @042-4,+ 50+' @0/,410 +03,2-405 (0-@;1+2,5 0510 @;1-/- 1040 3,5 5(*041052,+,210+<512,5/08*42-4,@041-J

C -=-152-4.-2-5(+,/-5/03=0+1,/ C

1 1 2 1 2

cos cos( )

( )

x l q l q q

y l senq l sen q q

= + +

)(*+,-!03,274(0-@;1+2,/0*4053,=74

2-414*,274051,=3020+0@-503@;1-/-@0/,410@,1+205/01+,458-+@,274D-@-(;40, 032*,310403,55(*041052,+,210+<512,5J

C

2,/,053,=745030,5-2,*45510@,/0+080+042,5-3/,+-C 5 .-5=30 +0.+05041,+ 3,5 1+,53,2-405 6 +-1,2-405 +03,1,5 041+0 3-5 /5141-5 053,=-405

C ,@,1+Ei 1

i

A

− _{+0.+05041,3,.-52746-+041,274+03,1,041+03-55510@,5,5-2,/-5}

,/-5053,=-4052-4502*1-5/03+-=-1

C 0.+05041,2741-1,3-.,+2,3/03,2,/04,240@'12,/03+-=-1J

1 2 0 0

3 1 2 3

1 2 3 4 5 0 0

6 1 2 3 4 5 6

A A A A

T A A A A A A A

=

= =

C ?5104 @;1-/-5 5510@'12-5 .,+, 51*,+ 3-5 5510@,5 /0 2--+/04,/,5 ,5-2,/-5 , 2,/,053,=746-=1040+3,2,/04,240@'12,/03+-=-1

0.+05041,274/03@-/03-%"&'%&'

• 0+@10 03 .,5- /0 *4 053,=74 ,3 5(*0410 @0/,410 1+,458-+@,2-405 ='52,5I

>*0/0.04/040?23*5,@0410/03,52,+,210+<512,52-451+*21,5/03+-=-1

• ,51+,458-+@,2-405='52,5>*0+03,2-4,4035510@,/0+080+042,/03030@041- 2-4035510@,/03030@041-i−15-4J

-1,274U,3+0/0/-+/030A0z_i₋₁ " +,53,274/,3-3,+(-/030A0z_i₋₁ +,53,274,,3-3,+(-/030A0x_i -1,274V,3+0/0/-+/030A0x_i

(

) (

)

1

1 1

, 0, 0, , 0, 0 ,

i

i i i

A T z θ T d T a T x α

−

(27)

%

0 0 1 0 0 0 1 0 0 1 0 0 0

0 0 0 1 0 0 0 1 0 0 0 0

0 0 1 0 0 0 1 0 0 1 0 0 0

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

0

0 0 0 1

i i i

i i i i

i i i

i i i i i i i

i i i

C S

S C C S

d S C

C C S S S C

S C C S C S

S C d

θ θ α θ θ α α α α θ α θ α θ α θ θ α θ α θ α θ α α − ª º ª º ª º ª º « » « » « » « » − « » « » « » « » = « » « » « » « » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ − ª º « » − « » = « » « ¬ ¼»

03,3(-+1@-/0 %"&'4-5/20>*0(

• *@0+,+ 3-5 053,=-405 2-@04E,4/- 2-4)9.+@0+ 053,=74 @73 /0 3, 2,/04,: 6

,2,=,4/-2-49H31@-053,=74@73:04*@0+,+'2-@-053,=74&,3,=,508A,/03 +-=-1

• *@0+,+ 2,/, ,+12*3,274 2-@04E,4/- .-+)93, 2-++05.-4/0410 ,3 .+@0+ (+,/- /0

3=0+1,/:6,2,=,4/-04

•

-2,3E,+030A0/02,/,,+12*3,274;51,05+-1,1,I030A050+'5*.+-.-0A0/0(+-

05.+5@'12,I50+'030A0,3-3,+(-/032*,350.+-/*2003/05.3,E,@041-• ,+,/0*,&)51*,+030A0E5-=+0030A0/03,,+12*3,274+)

• 1*,+03-+(04/035510@,/03,=,50

{ }

S0 042*,3>*0+.*41-/030A0z0-50A05x0 0y₀5051*,+'4/0@-/->*08-+@04*45510@,/0?1+7(+-2-4z₀

• ,+,/0),&)I51*,+035510@,

{ }

S_i 95-3/,+-,3053,=74:043,

• 410+502274/030A0E2-43,3<40,4-+@,32-@H4,EL6E,@=-50A05502-+1,504

50 51*,+<, WX 04 03 .*41- /0 2-+10 8*0504 .,+,303-5

_{{ }}

S_i 50 51*,+<, 04 3, ,+12*3,274+)

• ,+,/0),&)I51*,+?043,3<40,4-+@,32-@H4,zi−16E

• ,+,/0),&)I51*,+6/0@-/->*08-+@0*45510@,/0?1+7(+-2-4?6E

• 1*,+035510@,

{ }

Sn 04030?1+0@-/03+-=-1/0@-/->*0E42-42/,2-43,/+02274

/0z_n₋₁6?450,4-+@,3,z_n₋₁6E4

• =1040+U2-@-03'4(*3->*0D,6>*0(+,+041-+4-,zi−1.,+,>*0xi−1 6?>*0/04 .,+,303-5

• =1040+ / 2-@- 3, /51,42,I @0//, , 3- 3,+(- /0 zi−1I >*0 D,=+<, >*0 /05.3,E,+

{

Si−1

}

.,+,>*0xi6xi−1>*0/,504,340,/-5

• =1040+,2-@-3,/51,42,@0//,,3-3,+(-/0?I>*0,D-+,2-42/+<,2-4xi−1I>*0 D,=+<,>*0/05.3,E,+034*0-

{

Si−1

}

.,+,>*05*-+(042-42/0502-4

{ }

Si

• =1040+V2-@-03'4(*3->*0D,=+<,>*0(+,+041-+4-,?I>*0,D-+,2-42/+<,2-4

1

i

x₋ I.,+,>*0034*0-

{

S_i−₁

}

2-42/0501-1,3@04102-4

{ }

Si

• =1040+3,5@,1+205/01+,458-+@,274i 1

i

A

(28)

!

• =1040+ 3, @,1+E /0 1+,458-+@,274 >*0 +03,2-4, 03 5510@, /0 3, =,50 2-4 03 /03

0?1+0@-/03+-=-1 0 0 1 2 3 4 5 6 1 2 3 4 5 6

T = A = A A A A A A

• , @,1+E/0840 3, -+041,274 95*=@,1+E /0 +-1,274: 6 .-5274 95*=@,1+E /0

1+,53,274:/030?1+0@-I+080+/,5,3,=,50048*4274/03,52--+/04,/,5,+12*3,+05

)(*+,-&[email protected]/0240@'12,/+021,

%&

3.><,83?./03 .+-=30@, 240@'12- 40+5- 2-45510 04 042-41+,+ 3-5 ,3-+05 >*0 /0=04

,/-.1,+3,52--+/04,/,5,+12*3,+05/03+-=-1Iq=

_{

q q₁, ₂,...q_n

_}

I.,+,>*05*[email protected] -+041050(H4*4,/010+@4,/,3-2,3E,27405.,2,3,+05-3*2744-055510@'12,J0.04/0/03, 2-48(*+,274/03+-=-16.*0/040?51+5-3*[email protected]

• .-723452,001;1%

(

, , , , ,

)

qk = fk x y z α β γ

C -5=3/,//0+05-3*[email protected]+0,3

C -5=3/,//0423*++051+22-405>*0(,+,412043,@0A-+5-3*274 C -5=3/,//[email protected],2-405

C -50@.+00?510

• @8.;.6A,[email protected],68.6,.>6,0?1,5-16B3A70161C>J

(29)

"&

1

2 2 2

2 2 3 2 3 3 2 2 2 2 2

2 3 3 2 3 2 3 3 2 3 3 3 arctan 2 cos cos 2

sin 1 cos

1 cos arctan cos y x x y x

x y z

p q

p

r p p

r p l l l l q

p p p l l

q l l q q q q q § · = ¨ ¸ © ¹ = + = + + + + − − = = ± − §_± ₋ · ¨ ¸ = _¨ _¸ © ¹ )(*+,-9,:05-3*274.-+@;1-/-5(0-@;1+2-5 2 2 2 3 3

2 3 3 arctan sin arctan cos z x y q p p p l q

l l q

β α β α = − § · ¨ ¸ = ¨_± ₊ ¸ © ¹ § · = ¨ ¸ + © ¹ )(*+,-9=:05-3*274.-+@;1-/-5(0-@;1+2-5 0051,8-+@,50-=10403,5(*04108-+@*3,J 3 3 2 ₂ ₂

2 3 3 sin arctan arctan cos z x y l q p q

l l q

(30)

"

• 18032,6;,80156B.0912345D.9.A@5,1J

C 05.0A,+3,5,+,=305q_i048*4274/03,[email protected]/03-5021-+055I.I16/

1 1 2 2

0 1 2 1 2 3

1 3

1 2 1 1 2 1 2 1 1 2 3 1 2

1 2 1 1 2 1 2 1 1 2 3 0 0

2 3 2 2 1

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 1 0 0 1 0 0 0 1 1

0 0 0 1 0 0 0 1 0 0 0 1

0 0 0

0 0 0 1

C S C S

S C S C

A A A

l d

C C S C S C C S C S d S S

S C C S S S C C S S d

A T A

S C l

− ª º ª º ª º « » « » « » − « » « » « » = = = « » « ₋ » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ − − − − − ª º « » − − − « » = = = « » « » ¬ ¼ 1 2

2 0 2 3 2 1

0 0 0 1

S S

S C d C l

ª º « » « » « ₊ » « » ¬ ¼ )(*+,-"05-3*274.-+@,1+205/01+,458-+@,274 -4051-50-=10403,55(*04105@,1+205/01+,458-+@,274J

( )

1 2 0

1 2 3 1 ₂ 0 1

1 2 3 T A A A

A − T A A

=

1 1 2 2

1 3

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 1 0 0 1 0 0 0 1 1

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

x x x x

y y y y

z z z z

C S n o a p C S

S C n o a p S C

l n o a p d

− ª º ª º ª º ª º « » « » « » « » − « » « » «₋ » « » « » « » « ₋ » « » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼

2 2 2

0 3

0 1 0 0

0 0 0 1

C S S d

S C C d

− − ª º « » « » =_« _» − « » ¬ ¼ 3(04/-03030@041-9I:

1 1 0 tan 1

y

x y

x

p S p C p

p

θ § ·

− = =¨ ¸

(31)

""

( ) ( )

₁ 1 ₀ 1 ₂ 2 1 3 A − A − T = A

2 2 1 1

1

2 2 1 1 3

0 0 0 0 1 0 0 0

0 0 1 0 0 0 1 0 1 0 0

0 0 0 0 0 1 1

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

x x x x

y y y y

z z z z

C S C S n o a p

l n o a p

S C S C n o a p d

ª º ª º ª º ª º « » « » « » « » − − « » « » « »₌« » «₋ » « ₋ » « » « » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼

2 1 2 1 2 1 2

1 1

2 1 2 1 2 1 2 3

1 0 0 0

1 0 0 1 0 0

0 1 1

0 0 0 1 0 0 0 1 0 0 0 1

x x x x

y y y y

z z z z

C C C S S l S n o a p

S C n o a p

S C S S C l C n o a p d

− ª º ª º ª º « » « » « » − « » « »₌« » «₋ ₋ ₋ » « » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ 3(04/-03030@041-9I: 1 1 2 1 2 1 2 1 2

1

0 2 arctan x y

x y z

z

C p S p C C p C S p S p l S

p l θ + + + − = = − − 3(04/-03030@041-9I:

(

)

(

)

2 1 2 1 2 21 3 3 2 z 1 2 1 x 1 y

S C px S S py C pz C l d d C p l S C p S p

− − + − = = − − +

• ,612./-.235,9E832.J

C ,+,/010+@4,/-5+-=-152-4 (+,/-5/03=0+1,/

C 05-3*274 4/0.04/0410 /0 3-5 (+,/-5 /0 3=0+1,/ >*0 .-52-4,4 6 /0 3-5 >*0 -+041,4 0 5 4 6 0 6 m m r r

p O O

p p l z p O O

=

₌ ₋

=

6

; ; T; ; ; T

r x y z x y z

p =ª_¬p p p º_¼ z =ª_¬a a a º_¼

4

mx x x

m my y y

mz z z

p p l a

p p p l a

p p l a

(32)

"

%&

4 3, .-5274 .,+, 50(*+ *4 2-41-+4- , *4, 03-2/,/ 2-451,410I - , *4, 03-2/,/ .+0052+1,I4-5-1+-5/0=0@-52-4-20+3,+03,274041+03,03-2/,//03,D0++,@041,63,03-2/,/ /03,,+12*3,274405102,5-4-5-1+-5.-/0@-5/80+042,53,502*,2-4056".,+,-=1040+

) )(

sin(

sin ₁ ₁ ₂ ₁ ₂ 1 2

1 ⋅ ⋅ ⋅ ⋅ + + − ⋅ −

= a θ θ a θ θ θ θ

x 9:

) )(

cos(

cos ₁ ₁ ₂ ₁ ₂ 1 2

1 ⋅ ⋅ ⋅ ⋅ + + + ⋅

=a θ θ a θ θ θ θ

y

5,4/- *4, 4-1,274 /0 021-+ _¸¸

x 6 _¸¸

θ 4-5-1+-5 .-/0@-5 052+=+ 3,5 02*,2-405

2-@-» ¼ º « ¬ ª + + + + − + − − = ⋅ ) cos( ) cos( cos ) sin( ) sin( sin 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 θ θ θ θ θ θ θ θ θ θ a a a a a a x 9": ⋅ ⋅

=Jθ

x

, @,1+E Q /084/, 04 9": 05 33,@,/- 03 Q,2-=,4- /03 @,4.*3,/-+ 6 05 03 -=A01-8*4/,@041,3,/010+@4,+04,3(*4-5@,4.*3,/-+05

, /010+@4,274 /0 3, 03-2/,/ /0 ,+12*3,274 /05/0 3, 03-2/,/ /03 030@041- 84,3 05 2-420.1*,3@0410 [email protected] 54104/- >*0 3, +03,274 /0 3, 03-2/,/ 05 340,3 5< 3, 03-2/,/ /0

,+12*3,27405=*52,/,/05/03,03-2/,//03030@041-84,3<,Q,2-=,4-40+5-⋅ − ⋅

=J 1x

θ 9:

» ¼ º « ¬ ª » ¼ º « ¬ ª + − − + − − + + = » » ¼ º « « ¬ ª ⋅ ⋅ y x a a a a a a a

a cos cos( ) sin sin( )

) sin( ) cos( sin 1 2 1 2 1 1 2 1 2 1 1 2 1 2 2 1 2 2 2 1 2 1 θ θ θ θ θ θ θ θ θ θ θ θ θ _9:

,/010+@4,410/01Q/03Q,2-=,4-049":05a a₁, ₂, sinθ₂3Q,2-=,4-4-104040+5,I .-+ 2-45(*0410I 2*,4/- θ₂ =0I 04 2*6- 2,5- 03 @,4.*3,/-+ 05 4/2,/- .,+, 50(*+ *4,

2-48(*+,274 54(*3,+ , /010+@4,274 /0 2,/, 2-48(*+,274 54(*3,+ 05 @.-+1,410 .-+ ,+,5 +,E-40543,52-48(*+,2-40554(*3,+05D,6@-@041-5484105@,30505-55-44-+0,3E,=305K 05/02+I03030@041-84,3/03@,4.*3,/-+4-.*0/0@-0+500420+1,5/+022-40543-52,5-5 ,410+-+05 03 030@041- 84,3 4- .*0/0 041+,+ 3, /+02274 .,+,303, , , /05/0 *4, 2-48(*+,274 54(*3,+ ,5 2-48(*+,2-405 +03,2-4,/,5 5-4 1,@=;4 +03,2-4,/,5 .,+, 3, 8,31, /0 *42/,/ /0 5-3*2-405/03,240@'12,40+5,

(33)

"

%&

4+-=-1@,4.*3,/-+05='52,@0410*4/5.-51-/0.-5274,+,032-41+-3/0.-5274 4-5-1+-5 /0=0@-5 2-4-20+ 3,5 .+-.0/,/05 /0 3, /4'@2, /03 @,4.*3,/-+ .,+, 5,=0+ 2*,41, 8*0+E, 50 /0=0 0A0+20+ 04 03 .,+, .-/0+3- @-0+ -4 .-2, 8*0+E, 03 @,4.*3,/-+ 05 3041- 04 +0,22-4,+-4/0@,5,/,8*0+E,I03=+,E-.*0/02D-2,+2-43-5-=A01-5-.*0/0-523,+5-=+03, .-5274/050,/,0+,4/-3,502*,2-405/03,/4'@2,/03@-@041-/03+-=-1051-4-05*4, [email protected],+0,@0+02/,.,+,*4(+,44H@0+-/0(+,/-5/03=0+1,/64-340,3/,/05.+05041050403 5510@,

! !%

,10-+<,/02-41+-305*5,/,.,+,/50B,+,3(-+1@-5/02-41+-3.,+,3,0A02*274/01,+0,5 .+-(+,@,/,5 3 .+-=30@, /0 2-41+-3 /0 @-@041- 2-45510 04 03 ,51+0- 6 03 .+-=30@, /03 +02D,E- /03 /51*+=-I 2*,3 05 03 .+-=30@, /0 /010+@4,+ 3,5 041+,/,5 /0 2-41+-3 40205,+,5 .,+, 50(*+ - +,51+0,+ *4, 1+,6021-+, /050,/, >*0 1040 , 50+ .3,40,/, .-+ 03 @,4.*3,/-+I @041+,5 5@*31'40,@0410 +02D,E, 3-5 /51*+=-5 @0+02/-5 .-+ 3-5 08021-5 /0 3, /4'@2, 4- @-/03,/,I 2-@-I 8+2274 6 +*/- -5-1+-5 /01,33,@-5 03 .,1+74 /0 ,.+-02D,@041- /03 2-41+-3 /03 +-=-1 =,5,/-041;242,5/03/-@4-/03,8+02*042,

" "%'

4, 0E 03 @,4.*3,/-+ 104/0 ,32,4E,+ 3, 3-2,274 I 0510 /0=0 50(*+ 03 2-41-+4- @,410404/-*4,8*0+E,4-+@,32-451,410043,5*.0+820-420==30@0410I2-4-204/-3,3-2,274 /03-=A01-63,8-+@,/032-41-+4-I4-5-1+-5.-/0@-52,+(,+8*0+,05,1,+0,*5,4/-5-3-032-41+-3 /0.-5274

! "

4,(+,4/0.,+10/03,240@'12,/0+-=-1052-420+4/-2-403051,=302@041-/0,+-5 5510@,5 2--+/4,/-5 .,+, +0.+05041,+ 3, .-5274 6 -+041,274 /0 -=A01-5 +<(/-5 6 2-4 1+,458-+@,2-405 041+0 05-5 5510@,5 2--+/4,/-5 0+1,@0410I 3, (0-@01+<, /03 05.,2-1+/@045-4,36/0@-@041-5+<(/-5/[email protected],4/-*4.,.032041+,3041-/-53-5,5.021-5/0 @,4.*3,274 +-=712, 4 051, .,+10 4-5-1+-5 051*/,@-5 3, -.0+,274 6 +-1,274 /0 1+,53,274 0 41+-/*203,4-274/01+,458-+@,2-405D-@-(;40,5,51+,458-+@,2-405D-@-(;40,52-@=4,4 3,5 -.0+,2-405 /0 +-1,274 6 1+,53,274 /041+- /0 *4, [email protected] @,1+E /0 @*31.32,274 -5-1+-5 1,@=;4 4051(,@-5 3, 1+,458-+@,274 /0 03-2/,/ 6 ,2030+,2-405 041+0 3-5 5510@,5 2--+/4,/-5

(34)

"#

!

,8(*+,@*051+,*4-=A01-+<(/-032*,3*4@,+2-2--+/4,/-0 1

x 61z₁ 058A,/--5-1+-5/050,@-53,+03,274/02--+/04,/,5/0*4.*41-/040403@,+2-/02--+/04,/,5

0 1

x 6 1z₁ /0 . 04 *4 ,++0(3- 9- 4- +-/,/-: /0 @,+2- /0 +080+042, 0x₀6 0z₀

-404/-{

i j k0, 0, 0

}

4/2,03.,1+74-+1-4-+@,3=,5004

0 0

x 60z₀K,5<i₀, j k₀, ₀ 5-4*4021-+*41,+-041+0 3-50A05x y z₀, ₀, ₀+05.021,@0410@3,+@0410I.*0/0

{

i j k₁, ,₁ ₁

}

50+03.,1+74-+1-4-+@,3=,5004

0 1

x 6 1z₁ 41-4205 03 021-+ /05/0 03 -+(04 2-@H4 04 03 .*41-/ 04 03 -=A01- .*0/0 50+ +0.+05041,/-,@=-52-4+05.021-,0

0

x 60z₀-@-J 0 0 0 0 0 0

0 p i p j p k

p = _x + _y + _z 9#:

-2-4+05.021-,0 1

x 61z₁

2-@-1 1 1 1 1 1

1 p i p j p k

p = _x + _y + _z 9 :

)(*+,--1,274/0*4+-=-1

05/0 041-4205 p₀ 6 p₁ 5-4 +0.+05041,2-405 /0 ,3(H4 021-+ .I 3, +03,274 041+0 3,5 [email protected]/0.040350(*4/-@,+2-2--+/4,/-.*0/050+-=104/-2-@-5(*0 0 1 1 1 1 0 1 1 0 1 0 0 0 i k p i j p i i p i p i p p z o y x x + + = = = 9$: -5-1+-51040@-58-+@*3,55@3,+05.,+, y o

p 6 ₀ z p I2*,3055-4J 0 1 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 k k p k j p k i p p j k p j j p j i p p z o y x z z o y x y + + = + + = 9%: -5-1+-5.-/0@-5052+=+3,51+0502*,2-40521,/,5,410+-+@04102-4A*41,52-@-J 1 1 0 0 R p

(35)

" /-4/0 » » » ¼ º « « « ¬ ª = 0 1 ' 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 k k k j k i j k j j j i i k i j i i

R 9&:

,@,1+E/0?+0.+05041,3,@,1+E/01+,458-+@,274/05/03,52--+/04,/,5/0.2-4 +05.021-,3 @,+2- /0 2--+/04,/,50

1

x 61z₁ 2-4 +05.021-,3 @,+2-0x₀60z₀; ,5<I 5 /,@-5 *4 .*41- 0510 05 0?.+05,/- 04 2--+/04,/,50

1

x 61z₁ 2-@- . 041-4205R₀1p₁ +0.+05041,4 ,3(H4 021-+0?.+05,/-+03,1-0403@,+2-/02--+/04,/,50

0 x 60z₀ @3,+@04104-5-1+-5.-/0@-5052+=+ 1 0 0 1 0 0 1 0 0 1 0 1 1 1 i k p i j p i i p i p i p p z y x x + + = = = 9: 12I-048-+@,/0@,1+E 0 0 1 1 R p

p = 9":

/-4/0 » » » ¼ º « « « ¬ ª = 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 k k k j k i j k j j j i i k i j i i

R 9:

5<3,@,1+E 0 1

R +0.+05041,3,40+5,/03,1+,458-+@,274R₀1 40@=,+(-03.+-/*21-410+-+052-4@*1,1-I0Ii j_{0 0} = j i_{0 0}

I012I4-5-1+-50@-5051-( )

( )

T

R R

R₁0 = 1₀ −1 = 1₀ 9:

,3@,1+E 0

1

R 2*6,40+5,055*1+,45.*051,05/02+5*-+1-(-4,33021-+2-3*@4,R1₀ 05/0*4/,/3,+(,6@*1*,@0410-+1-(-4,3

!

02-+/,4/- 3, @,1+E 1 0

R 04 3, 02*,274 9": +0.+05041, *4, 1+,458-+@,274 +-1,2-4,3 041+0 3-5 @,+2-5 0

0

x 6 0z₀ 6 0x₁ 61z₁ *.-40@-5 4-5-1+-5 5*@,+ ,D-+, *4 10+20+ @,+2-2--+/04,/- 0

2

x 6 2z₂ ,5-2,/- , 3-5 @,+2-5 0x₀ 6 0z₀ 6 0x₁ 6 1z₁ .-+ 3, 1+,458-+@,274 +-1,2-4,34.*41-/,/-..*0/0041-420550++0.+05041,/-041+058-+@,5J.&I.6."043-51+05 @,+2-5,+03,274041+005,5+0.+05041,2-405/0.05J

1 1 0 0 R p

(36)

"$ 2

2 0 0 R p

p = 9 :

2 2 1 1 R p

p = 9$:

-4/02,/, j i

R 05*4,@,1+E/0+-1,274-1,05,R₀16R₀2+0.+05041,3,+-1,274+03,1, /03-50A05R& &Y&I@041+,5R₁2+0.+05041,*4,+-1,274+03,1,/03@,+2-0x₁ 61z₁ *511*604/-9$:/041+-/09#:.+-/*2@-5 2 2 1 1 0 0 R R p

p = 9%:

-@.,+,4/-9 :69%:4-5-1+-51040@-53,/041/,/ 2 1 1 0 2 0 R R

R = 9!:

,02*,274 9!: 05 3, 306 /0 [email protected] .,+, 3,5 1+,458-+@,2-405 +-1,2-4,305 4 0510051,/-I04-+/04,3,1+,458-+@,274/02--+/04,/,5/0*4.*41-./05/005,+0.+05041,274 ."0403@,+2-0x₂ 62z₂,051,5+0.+05041,2-405.&0403@,+2-0x₀ 60z₀I4-5-1+-5/,@-53, .+@0+, 1+,458-+@,274 , 051,5 2--+/04,/,5 . 04 03 @,+2- 0x₁ 6 1z₁ *5,4/- R₁2 6 3, 1+,458-+@,274.6.&*5,4/-R1₀

-5-1+-5.-/0@-5410+.+01,+3,02*,2749%:2-@-5(*0*.-4(,@-542,3@0410>*0 1+05/03-5@,+2-52--+/4,/-52-42/04*051+-.+@0+(+-/03@,+2-0

1

x 61z₁+03,1-,&?&6&E& ,2-+/,4/- 3, 1+,458-+@,274 1

0

R 41-4205I 2-4 3-5 @,+2-5 2-42/041050x₁ 61z₁ 60x₂ 62z₂I 4-5-1+-5(+,@-50

2

x 62z₂+03,1-,0x₁ 61z₁,2-+/,/-043,1+,458-+@,274R₁2 42,/,2,5-4-5-1+-533,@,@-503@,+2-+03,1-042*6,5+-1,2-405-2*++040403@,+2-,21*,3

!

,5 +-1,2-405 4- 50@.+0 50 /[email protected],4 5-=+0 03 0A0 2--+/4,/- .+42.,3 -5-1+-5 051,@-5 , @04*/- 410+05,/-5 04 *4, +-1,274 5-=+0 03 0A0 ,+=1+,+- 04 03 05.,2- -+ 2-45(*0410.-40@-5k=

(

k k k_x, _y, _z

)

TI0?.+05,/-0403@,+2-0

0

x 60z₀ I05*4021-+*41,+-/084/-04030A0-5-1+-5/050,@-53,/0+,/,/03,@,1+E/0+-1,274R_k_,_θ+0.+05041,/-*4, +-1,274/0θ(+,/-55-=+00500A0

G,6,+,58-+@,5043,2*,33,@,1+ER_k_,_θ .*0/050+/0+,/,Z*E'53,*4,[email protected]+@, /053,+-1,274/0021-+N/041+-/0*4-/03-50A052--+/4,/-5I/(,@-5z₀.-+θ 684,3@041050 (+, N D,2, ,1+'5 , 3, .-5274 -+(4,3 05/0 041-4205 1-/,5 3,5 +-1,2-405 5-4 /[email protected],/,5 +03,1,5,*4@,+2-8A-0

0

x 60z₀3,@,1+ER_k_,_θ α β θ β α θ = , , , ,− ,−

, z y z y z

k R R R R R

R 9"&:

(37)

"% 03,8(*+,I/,/->*0N05*4021-+*41,+-I4-5-1+-50@-5>*0 2 2 sin y x y k k k + =

α 9":

2 2 cos y x x k k k + = α 2 2 sinβ = k_x +k_y

z k = β cos » » » ¼ º « « « ¬ ª + + − − + + + − + = θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ c v k s k v k k s k v k k s k v k k c v k s k v k k s k v k k s k v k k c v k R z x z y y y x x z y y z y x y z x z y x x k 2 2 2

, 9"":

-4/0v_θ[0+5θ[1−c_θ

!

-54*00030@041-5r_ij04*4,1+,458-+@,274+-1,2-4,3(040+,32-@-049&:4-5-4 2,41/,/054/0.04/041050+1,@0410*42*0+.-+<(/-.-500,3-5*@-1+05(+,/-5/03=0+1,/ +-1,2-4,305 6 , 3- 5*@- 1+05 2,41/,/05 5-4 +0>*0+/,5 , 05.0282,+ 05, -+041,274 4 051, 5022744-5-1+-5/0+,@-51+058-+@,5042*6,+-1,274,+=1+,+,.*0/050++0.+05041,/,*5,4/-5-3-1+052,41/,/054/0.04/04105,.+@0+,053,+0.+05041,2740A0T'4(*3-,50(*4/,053, +0.+05041,274/0'4(*3-/0*30+63,10+20+,053,+0.+05041,274+-33L.12DL6,S

! (&

4,@,1+E/0+-1,274∈&9:.*0/050@.+050++0.+05041,/-.-+*4,[email protected]+-1,274 θ , k R

R= 9":

-4/0N05*4021-+*41,+-/084/-.-+030A0/0+-1,274I6θ0503'4(*3-/0+-1,274 5-=+0 N , 02*,274 9": 05 33,@,/, 3, +0.+05041,274 0A0L'4(*3- /0 ,4/- *4, @,1+E /0 +-1,274,+=1+,+,[email protected]r_ij:I03'4(*3-0>*,30410θ6030A00>*,30410N5-4/,/-5 .-+3,50?.+05-405

(38)

"! ¸ ¹ · ¨ © § + + − = ¸ ¹ · ¨ © § − = − − 2 1 cos 2 1 ) ( cos 33 22 11 1 1 r r r R Tr θ 9": -4/0+/04-1,031+,E-/0I6 » » » ¼ º « « « ¬ ª − − − = 12 21 31 13 23 32 cos 2 1 r r r r r r k

θ 9"#:

,+0.+05041,2740A0T'4(*3-4-05H42,/05/0*4,+-1,274/0θ(+,/-55-=+0LN>*0 051,(*,35-=+0*4,+-1,274/0θ5-=+0NI051-05

θ θ = − ,−

, k

k R

R 9" :

θ

[&041-4205053,@,1+E/041/,/6030A0/0+-1,274054/084/-! &

4 @;1-/- @'5 2-@H4 /0 05.0282,+ *4, @,1+E /0 3, +-1,274 04 1;+@4-5 /0 1+05 2,41/,/05 4/0.04/04105 05 *5,+ 03 33,@,/- '4(*3- /0 *30+ -45/0+,4/- -1+, 0E 03 @,+2-2--+/4,/-8A-0

0

x 60z₀603@,+2-(+,/-0x₁ 61z₁

-5-1+-5.-/[email protected],+3,-+041,274/03@,+2-+03,1-0 1

x 61z₁/03@,+2-0x₀ 6 0

0

z .-+1+05'4(*3-5

(

θ,φ,ψ

)

I@-51+,/-2-@-'4(*3-/0*30+I6-=104/-.-+1+05(+-55*205-5

2-@-5(*04J03.+@0+(+-5-=+0030A0E.-+03'4(*3-φ35(*0410(+-5-=+0030A0,21*,3.-+03 '4(*3-θ 3 (+- 84,3 5-=+0 03 0A0 E ,21*,3 .-+ 03 '4(*3-ψ 4 1;+@4-5 /0 3, @,1+E ='52, +-1,2-4,303+05*31,/-/03,1+,458-+@,274+-1,2-4,3 1 0 R .*0/050+(040+,/,2-@-03.+-/*21-/0J ψ θ φ , , , 1

0 Rz Ry Rz

R = 9"$:

» » » ¼ º « « « ¬ ª − » » » ¼ º « « « ¬ ª − » » » ¼ º « « « ¬ ª − = 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1

0 ψ ψ

(39)

&

! )*&

4, @,1+E /0 +-1,274 .*0/0 50+ /052+1- 2-@- *4 .+-/*21- /0 +-1,2-405 5*205,5 5-=+0030A02--+/4,/-.+42.,3?&I6&6E&1-@,/-04*4-+/0405.0282-5,5+-1,2-405/08404 3-5'4(*3-5+-33I.12D66,SI/0/-4/04-5-1+-51,@=;4/04-1,@-5φ,θ,ψ

-5-1+-505.0282,@-50-+/04/03,+-1,2742-@-?L6LEI04-1+,5.,3,=+,5I.+@0+-*4, /05,2745-=+0030A0?&,1+,;5/0*4'4(*3-ψ I041-4205504234,5-=+0030A06&

04*4'4(*3-θI 6 84,3@0410 *4 (+- 058;+2- 5-=+0 03 0A0 E& 04 *4 '4(*3-φ 05/0 041-4205 3,5 +-1,2-405 5*205,55-4+03,1,5,*4@,+2-8A-I03+05*31,/-/03,@,1+E/01+,458-+@,27405/,/-.-+ ψ θ φ , , , 1

0 Rz Ry Rx

R = » » » ¼ º « « « ¬ ª − » » » ¼ º « « « ¬ ª − » » » ¼ º « « « ¬ ª − = ψ ψ ψ ψ θ θ θ θ φ φ φ φ c s s c c s s c c s s c 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 » » » ¼ º « « « ¬ ª − + − + + + − = ψ θ ψ θ θ ψ θ φ ψ φ ψ θ φ ψ φ θ φ ψ θ φ ψ φ ψ θ φ ψ φ θ φ c c s c s c s s s c s s s c c c s c s c s s s s c c s c c 9"%: -+5*.*051-I043*(,+/06,SL.12DL+-33+03,1-,03@,+2-8A-I4-5-1+-5.-/0@-5041-4205 410+.+01,+ 3, 21,/, 1+,458-+@,274 +-33L.12DL6,SI 04 03 -+/04I 2,/, 1-@, 2-4 +05.021- ,3 @,+2-,21*,33+05*31,/-84,305,3(*4,@,1+E9"%: " & "

,5 1+,45@5-405 5-4 030@041-5 042,+(,/-5 /0 1+,45@1+ 03 @-@041- /05/0 3-5 030@041-5 @-1-+05 D,51, 3,5 ,+12*3,2-405 6I 0041*,3@0410I +0,3E,+ *4, 2-40+574 /03 @5@-,5 1+,45@5-405 1040 3, 84,3/,/ /0 ,2030+,+ 03 @-@041- 04 03 0?1+0@- 6 ,3 @5@- [email protected]+0/*2+3,40+2,/03@-@041-3,51+,45@5-4055-4.,+05051'12-5/0.04/04/+021,@0410/0 3,/51,42,,3,5@,5,5I,/0@'5/0>*010404*4,20+2,@041-/03-5030@041-5@-1-+05,3,=,50 /03+-=-1

,5 1+,45@5-405 10404 3,5 .+-.0/,/05 /0 50+ /0 1,@,B- 6 .05- +0/*2/-I 10404 (+,4 +04/@041-I4-,8021,4,3@-@041-610403,(+,4041,A,/08*42-4,+2-414*,@0410,*4.,+ 030,/-51-50@*051+,043,8(*+,#

(40)

, 1+,45@574 /0 *4 @-@041- .*0/0 +0,3E,+50 .-+ @0/- /0 /80+04105 030@041-5 3-5 2*,3055-40?.*051-5043,8(*+,

)(*+,- 510@,5/01+,45@574

2-414*,274 50 0?.32,+,4 3-5 /80+04105 5510@,5 /0 1+,45@574 >*0 0?5104 .,+, *4 +-=-1

" +

83-3;1;

0 [email protected], .,+, 1+,45@1+ *4 @-@041- (+,1-+- 041+0 /-5 0A05 /51,4105 .0+@104/-,*@041,+I /5@4*+ - @,41040+ 3, 03-2/,/ /0(+- /03 0A0 2-4/*21-+I ,3 [email protected] >*0 @,41040+-40+1+ 03 5041/- /0 (+- /0 3-5 0A05 510 @02,45@- 05 @*6 [email protected],/- 04 ,.,+,1-5 03021+-/-@;512-593,,/-+,5I3,,,A33,5:I03021+742-59,.,+,1-5/0</0-6,*/-:604,3(*4-5 @02,45@-5/03-5@-1-+0590413,/-+I,310+4,/-+I=-@=,/0,(*,:

-+@,3@04103-50A0510404>*050+.,+,303-5I.0+-035510@,1,@=;4.*0/[email protected],+50 2-40A05>*0502+*E,404'4(*3-5480+-+05-(*,305,!&\51-50.*0/0,.+02,+043,8(*+, $

)(*+,-$-30,C2-++0,

(41)

"

,6203/2345

3 @*31.32,/-+ /0 03-2/,/ .-+ .-30,5 @'5 030@041,3 >*0 .*0/0 2-451+*+50 [email protected],I ,3 @04-5I 3-5 5(*04105 -.0+,/-+05J /-5,<,692-4/*21-+ 6 2-4/*2/-:I /-5/.-,16 B3<16/0 2-++0, 92-4/*21-+,62-4/*2/,:I*4,2.00,16*4,>16,5-=+03,>*08A,+1-/-032-4A*41-K,1-/-033-5030 .*0/04 ,B,/+ -1+-5 -.0+,/-+05 2-@-/.-,161045-+,5 - 3-2,5 2*6, 84,3/,/ 05 @0A-+,+ 03 [email protected]+1,@041-/035510@,

,*13/,//02,/,-.0+,/-+053,5(*0410J

• 3,<,2.5;728.005030A0>*0104003@-@041->*0>*0+0@-51+,53,/,+-1+,458-+@,+

9043,@,6-+<,/03-52,5-505030A0/03@-1-+:

• 3,<,2.5;723;.05030A0>*01040@-5>*0@-0+

• .-,12.5;728.01053,>*0051'*4/,,30A02-4/*21-+

• .-,12.5;723;1

053,>*0051'*4/,,30A02-4/*2/-• ,2.00,105*4,+-830?=30>*0,=+,E,,@=,5.-30,561+,45@1003@-@041-/0*4,,

-1+,

•

G,6>*0-=50+,+>*03-5/-51+,@-5/03,2-++0,4-50042*041+,45-.-+1,4/-03@5@-058*0+E-/0104574J*4-/0033-550042*041+,=-@=0,/-983-A-:@041+,5>*003-1+-051' 1-1,3@0410 1045- /0.04/04/- /03 5041/- /0 (+- /0 3, .-30, 2-4/*21-+, 904 3, 8(*+, 3, 5022745*.0+-+051'83-A-@041+,5>*003480+-+051,1045-:

• ,>16,053,042,+(,/,/05*A01,+,@=-50A056@,41040+3-5043,.-5274,/02*,/,4

,3(*4,5 @'>*4,5 0510 .0+,/+ /5.40 /0 *4 @02,45@ >*0 .0+@10 ,*@041,+ -/5@4*+3,/51,42,041+03-50A05.,+,.-/0+1045,+@'5-@04-53,2-++0,

)(*+,-%-30,C2-++0,"

"

83-3;1;

(42)

,6203/2345

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