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(1)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.

(2) I.

(3) II.

(4) III.

(5) IV.

(6) V.

(7) VI.

(8) VII.

(9) VIII.

(10) IX.

(11) 1.

(12) 2.

(13) 3.

(14) 4.

(15) 5.

(16) 6.

(17) 7.

(18) 8.

(19) 9.

(20) 10.

(21) 11.

(22) ❖. ❖. 12.

(23) ❖. ❖. ❖. ❖. ❖. 13.

(24) 14.

(25) 15.

(26) 16.

(27) (𝑥, 𝑦, 𝑧). 𝒂 𝑹 17.

(28) 𝒙. 𝑹𝑇 𝑹 = 𝑬 𝑬. Φ. 𝒏 𝒙=𝑹. ⃗⃗⃗⃗⃗ 𝑂𝐴 𝒙 ⃗⃗⃗⃗⃗ 𝑂𝑋 ⃗⃗⃗⃗⃗⃗⃗ 𝑂𝑂´. 𝒙 18.

(29) ⃗⃗⃗⃗⃗ 𝑂𝐴 𝒙. ⃗⃗⃗⃗⃗ 𝑂𝑋. ⃗⃗⃗⃗⃗⃗⃗ 𝑂𝑂’. ⃗⃗⃗⃗⃗⃗ 𝑂’𝐴. ⃗⃗⃗⃗⃗⃗⃗ 𝑂𝑂’. ⃗⃗⃗⃗⃗⃗ 𝑂’𝑋. ⃗⃗⃗⃗⃗⃗⃗ 𝑂𝑂´. ⃗⃗⃗⃗⃗⃗⃗ 𝑂𝑂´. 𝒏𝒏𝑻. ⃗⃗⃗⃗⃗⃗ 𝑂’𝐴 ⃗⃗⃗⃗⃗⃗ 𝑂’𝐴. 𝒏𝒏𝑻. 𝒏𝒏𝑻. ⃗⃗⃗⃗⃗⃗ 𝑂’𝑋. ⃗⃗⃗⃗⃗⃗ 𝑂’𝐴. ⃗⃗⃗⃗⃗⃗⃗ 𝑂’𝐵 ⃗⃗⃗⃗⃗⃗ 𝑂’𝑋. ⃗⃗⃗⃗⃗⃗ 𝑂’𝐴. ⃗⃗⃗⃗⃗⃗⃗ 𝑂’𝐵. ⃗⃗⃗⃗⃗⃗⃗ 𝑂’𝐵. ⃗⃗⃗⃗⃗⃗ 𝑂’𝐴 ⃗⃗⃗⃗⃗⃗⃗ 𝑂’𝐵. ⃗⃗⃗⃗⃗⃗ × 𝑂’𝐴. 𝒏𝒏𝑻. ×. 19. ×.

(30) ⃗⃗⃗⃗⃗⃗ 𝑂’𝑋. 𝒏𝒏𝑻. 𝒏𝒏𝑻. 𝒙. 𝑹. 𝑹𝑻. ×. 𝒏𝒏𝑻. 𝒏𝒏𝑻. ×. 𝒏𝒏𝑻. 𝒏𝒏𝑻. ×. 𝒏𝒏𝑻. ×. 𝑋𝑌𝑍. 𝑋𝑌𝑍. 𝜙 (0 ≤ 𝜙 < 2𝜋). 𝑍 𝑍𝑌. 𝜃 (0 ≤ 𝜃 < 𝜋) 20. 𝑋.

(31) 𝑋𝑥2 𝑥3 𝜓 (0 ≤ 𝜓 < 2𝜋). 𝑥3. 𝑥1 𝑥2 𝑥3. 𝑥1 𝑥2 𝑥3. 𝜙, 𝜃 𝑦 𝜓. 𝑋𝑌𝑍. 𝑹 = 𝑹𝟑𝒛 (𝜓)𝑹2𝑥 (𝜃)𝑹𝟏𝒛 (𝜙). Φ= 𝜙. 𝒏 = [0 0 1]𝑻. 21.

(32) 𝑐𝑜𝑠𝜙 𝑹𝟏𝒛 = (𝑠𝑒𝑛𝜙 0. −𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜙 0. 0 0) 1 𝒏 = [1 0 0]𝑻. 𝑹2𝑥 𝑂𝑋. Φ= 𝜃. 𝑹𝑥. 1 0 𝑹2𝑥 = (0 𝑐𝑜𝑠𝜃 0 𝑠𝑒𝑛𝜃. 0 −𝑠𝑒𝑛𝜃) 𝑐𝑜𝑠𝜃 𝒏 = [0 0 1]𝑻 𝜙. 𝑐𝑜𝑠𝜓 𝑹𝟑𝒛 = (𝑠𝑒𝑛𝜓 0. −𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓 0. 𝜓. 0 0) 1. 𝑹. 𝑐𝑜𝑠𝜓 𝑹 = (𝑠𝑒𝑛𝜓 0. −𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜓 0. 0 1 0 ) (0 1 0. 0 𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜃. 22. 𝑐𝑜𝑠𝜙 0 −𝑠𝑒𝑛𝜃) (𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜃 0. −𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜙 0. 0 0) 1.

(33) 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜃 𝑹 = (𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 + 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓. 𝑹1. −𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓. 𝑂𝑍. 𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜙 −𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜙) 𝑐𝑜𝑠𝜃. [1 0 0]𝑇. 𝜙. [cos𝜙 sin𝜙 0]𝑇 𝜃. 𝑂𝑍. 𝜓 [cos𝜓 sin𝜓 0]𝑇. 23. 𝜃. [cos𝜙 sin𝜙 0]𝑇.

(34) 24.

(35) 25.

(36) 𝐶2. {𝑃𝑘 }1𝑁. {𝜌𝑘 , 𝜃𝑘 }1𝑁. 𝛼 𝑙𝑣. 𝑙ℎ. 𝑘 = 1, … , 𝑁 𝜋. 𝜃1 = 𝛼, 𝜃𝑁 = 2 − 𝛽 𝑘−1 𝜋. 𝜃𝑘 = 𝛼 + 𝑁−1 [ 2 − (𝛼 + 𝛽)] 𝜌1 = √𝑙𝑣 2 + 𝑎2 𝜌𝑁 = √𝑙ℎ 2 + 𝑏 2 𝑁. 𝑁−2. 𝜌 𝐶2. 𝑃1. 90° − 𝛼. 𝑂𝑃1. 𝑃𝑁. 90° − 𝛽. 𝑂𝑃𝑁. 𝑃1 𝛾. 𝑃𝑁 𝜌(𝜃). tan 𝛾 = 𝜌′(𝜃) 𝜅. 26. 𝛽.

(37) 𝜅=. 𝜌1 𝜌1 ′. 𝜌2 +2(𝜌′)2 −𝜌𝜌′′. 𝜋. = tan( 2 − 𝛼). 𝜌𝑁 𝜌𝑁. 3 (𝜌2 +(𝜌′)2 ) ⁄2. 𝜋. = tan( 2 + 𝛽) ′. 𝜌1 2 + 2(𝜌1 ′)2 − 𝜌1 𝜌1 ′′ = 0 𝜌𝑁 2 + 2(𝜌𝑁 ′)2 − 𝜌𝑁 𝜌𝑁 ′′ = 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. 27. 𝑥𝑖 [𝑚𝑚]. 𝑦𝑖 [𝑚𝑚]. 𝜌𝑖 [𝑚𝑚]. 𝜅[𝑚𝑚].

(38) ̂𝒊 𝑖 = 1,2,3 𝒂. ̂𝒊 𝒃. 𝛼𝑖 𝛼𝑖 28.

(39) 𝒄̂𝒊. 𝛼𝑖. 𝑥𝑦𝑧. ̂𝟏 𝒂 ̂𝟐 𝑦 𝒂 ̂𝟑 𝒂 ̂𝒊 𝒃 ̂𝒊 ̂𝒊 , 𝒃 𝒂 𝒄̂𝒊. 29.

(40) ̂𝟏 𝒂 ̂𝟐 𝑦 𝒂 ̂𝟑 𝒂. ̂𝟏 = [1 0 0 ]𝑇 𝒂 ̂𝟐 = [0 1 0 ]𝑇 𝒂 ̂𝟑 = [0 0 1 ]𝑇 𝒂. 𝑇 ̂𝟏 𝒂 ̂𝟏 𝒃 ̂𝟐 𝒂. 𝛼1. ̂𝟑 𝒂. ̂𝟑 𝒂. ̂ 𝟏 = [0 − 𝑠𝑖𝑛𝛼1 𝑐𝑜𝑠𝛼1 ]𝑇 𝒃 ̂𝟐 𝒂 ̂𝟐 𝒃. 𝛼2. ̂𝟏 𝒂. ̂𝟏 𝒂. ̂𝟑. 𝒂. ̂ 𝟐 = [𝑐𝑜𝑠𝛼2 0 − 𝑠𝑖𝑛𝛼2 ]𝑇 𝒃 ̂𝟑 𝒂 ̂𝟑 𝒃. 𝛼3. ̂𝟐 𝒂. ̂𝟏 𝒂. 30.

(41) ̂𝟐 𝒂. ̂ 𝟑 = [ −𝑠𝑖𝑛𝛼3 𝑐𝑜𝑠𝛼3 0]𝑇 𝒃 ̂𝟏 𝒃. ̂𝟏 𝒂. ̂𝟏 𝒃 𝒄̂𝟏. ̂𝟏 𝒃 ̂𝟐 𝒃. ̂𝟐 𝒂. ̂𝟐 𝒃 𝒄̂𝟐. ̂𝟐 𝒃 ̂𝟑 𝒃. ̂𝟑 𝒂. ̂𝟑 𝒃 𝒄̂𝟑. ̂𝟑 𝒃. ̂𝟏 𝒃. ̂𝟑 𝒂. 𝑧. 𝒄̂∗𝟏 = [0 − 1 0]𝑻. ̂𝟐 𝒃 ̂𝟏 𝒂. 𝒄̂𝟐. 𝑦 𝒄̂∗𝟐 = [0 0 − 1]𝑻. 31. 𝒄̂𝟏.

(42) ̂𝟑 𝒃. ̂𝟐 𝒂. 𝒄̂𝟑. 𝑥 𝒄̂∗𝟑 = [−1 0 0]𝑻. 𝑧𝑦𝑥 𝑥 𝒄̂∗𝒊. 𝑧. 𝑹 = 𝑹𝟏𝒛 (𝜙)𝑹2𝑦 (𝜃)𝑹𝟑𝒙 (𝜓) 𝑹𝟏𝒛 (𝜙). 𝑹2𝑦 (𝜃). 32. 𝑦 𝒄̂𝒊.

(43) 𝒏 = [0 1 0]𝑻. Φ= 𝜃. 𝑹2𝑦 (𝜃) = (. 𝑐𝑜𝑠𝜃 0 −𝑠𝑖𝑛𝜃. 0 𝑠𝑒𝑛𝜃 1 0 ) 0 𝑐𝑜𝑠𝜃. 𝑹𝟑𝒙 (𝜓). 𝜃. 𝜓 1 0 𝑹𝟑𝒙 (𝜓) = (0 𝑐𝑜𝑠𝜓 0 𝑠𝑖𝑛𝜓. 𝑐𝑜𝑠𝜙 𝑹 = (𝑠𝑒𝑛𝜙 0. −𝑠𝑒𝑛𝜙 𝑐𝑜𝑠𝜙 0. 0 𝑐𝑜𝑠𝜃 ) ( 0 0 1 −𝑠𝑖𝑛𝜃. 0 −𝑠𝑒𝑛𝜓) 𝑐𝑜𝑠𝜓. 0 0 𝑠𝑒𝑛𝜃 1 0 𝑐𝑜𝑠𝜓 ) ( 1 0 0 𝑐𝑜𝑠𝜃 0 𝑠𝑖𝑛𝜓 𝜙, 𝜃. 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃 𝑹 = (𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃. 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓 𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓. 33. 0 −𝑠𝑒𝑛𝜓) 𝑐𝑜𝑠𝜓 𝜓. 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 + 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓) 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓.

(44) 𝒄̂𝒊 𝑖 = 1,2,3. 𝒄̂𝟏 =. 𝑹𝒄̂∗𝟏. 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 = (−𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓) −𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓. −𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 𝒄̂𝟐 = 𝑹𝒄̂∗𝟐 = ( 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 ) −𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓. 𝒄̂𝟑 =. 𝒄̂𝒊. 𝑹𝒄̂∗𝟑. −𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃 = (−𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃 ) 𝑠𝑒𝑛𝜃. ̂𝒊 𝒃. ̂ 𝒊 = 0, 𝒄̂𝒊 𝑻 𝒃. 𝑖 = 1,2,3.. (𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓)𝑠𝑒𝑛𝛼1 − 𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓𝑐𝑜𝑠𝛼1 = 0 (−𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓)𝑐𝑜𝑠𝛼2 + 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓𝑠𝑒𝑛𝛼2 = 0 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝛼3 − 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼3 = 0. 34.

(45) tan 𝛼1 = tan 𝛼2 =. 𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓+𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓+𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓. tan 𝛼3 = 𝑡𝑎𝑛𝜙. −𝒄̂𝟑. −𝒄̂𝟏. −𝒄̂𝟐. 𝒄̂𝒊. 𝒄̂13 tan 𝛼1 = 2 𝒄̂1. 𝒄̂12 tan 𝛼2 = 3 𝒄̂2. 𝒄̂𝑘𝑖. 𝒄̂23 tan 𝛼3 = 1 𝒄̂3. 𝒄̂𝒊. 𝑐𝑜𝑠𝜃𝑠𝑒𝑛(𝛼3 − 𝜙) = 0. 𝑐𝑜𝑠𝜃 = 0 𝑠𝑒𝑛(𝛼3 − 𝜙) = 0. 35.

(46) 𝜃 𝜃=. 𝜋 2. 𝜋. 𝜃 = −2. 𝜃=. 𝜋 2. 𝛼1 𝑐𝑜𝑠(𝜙 − 𝜓) = 0 𝜋. 𝜃 = −2. 𝑐𝑜𝑠(𝜙 + 𝜓) = 0. 𝜃=. 𝜋 2. 0 −𝑠𝑒𝑛(𝜙 − 𝜓) 𝑐𝑜𝑠(𝜙 − 𝜓) 𝑹=( 0 𝑐𝑜𝑠(𝜙 − 𝜓) 𝑠𝑒𝑛(𝜙 − 𝜓)) −1 0 0. 0 −1 0 𝑹𝟏 = ( 0 0 1) −1 0 0 0 1 0 𝑹𝟐 = ( 0 0 −1) −1 0 0. 36.

(47) 𝜋. 𝜃 = −2. 0 𝑹 = (0 1. −𝑠𝑒𝑛(𝜙 + 𝜓) 𝑐𝑜𝑠(𝜙 + 𝜓) 0. 0 𝑹 𝟑 = (0 1. −𝑐𝑜𝑠(𝜙 + 𝜓) −𝑠𝑒𝑛(𝜙 + 𝜓)) 0. −1 0 0 −1) 0 0. 0 1 0 𝑹 𝟒 = (0 0 1 ) 1 0 0. 𝒄𝟏 =. 𝑹𝟏 𝒄∗𝟏. 1 = [0] 0. −1 𝒄𝟏 = 𝑹𝟐 𝒄∗𝟏 = [ 0 ] 0. 𝒄𝟐 =. 𝑹𝟏 𝒄∗𝟐. 0 = [−1] 0. 0 𝒄𝟐 = 𝑹𝟐 𝒄𝟐∗ = [1] 0. 37. 𝒄𝟑 =. 𝑹𝟏 𝒄𝟑∗. 0 = [0 ] 1. 0 𝒄𝟑 = 𝑹𝟐 𝒄𝟑∗ = [0] 1.

(48) 𝒄𝟏 =. 𝒄𝟏 =. 𝑹𝟑 𝒄∗𝟏. 𝑹𝟒 𝒄∗𝟏. 1 = [0] 0. 𝑹𝟑 𝒄𝟐∗. 𝒄𝟐 =. −1 =[ 0 ] 0. 𝒄𝟐 =. 𝑹𝟒 𝒄∗𝟐. 0 = [1 ] 0. 0 = [−1] 0. 𝒄𝟑 =. 𝑹𝟑 𝒄∗𝟑. 0 =[ 0 ] −1. 𝒄𝟑 =. 𝑹𝟒 𝒄∗𝟑. 0 =[ 0 ] −1. 𝑠𝑒𝑛(𝛼3 − 𝜙) = 0. 𝜙 = 𝛼3. 𝜙. 𝜙 = 𝛼3 ± 𝜋. 𝜙 = 𝛼3. (𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼1 )𝑠𝑒𝑛𝜓 + 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑐𝑜𝑠𝜓 = 0 −𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑠𝑒𝑛𝜓 + (𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛼2 − 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝜃)𝑐𝑜𝑠𝜓 = 0. {. 𝑝11 𝑠𝑒𝑛𝜓 + 𝑝12 𝑐𝑜𝑠𝜓 = 0 𝑝21 𝑠𝑒𝑛𝜓 + 𝑝22 𝑐𝑜𝑠𝜓 = 0. 38.

(49) 𝑝11 = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼1 𝑝21 = −𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑝12 = 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑝22 = 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛼2 − 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝜃 𝑠𝑒𝑛𝜓. 𝑝11 𝑝22 − 𝑝21 𝑝12 = 0. 𝑐𝑜𝑠𝜃(𝐴𝑠𝑒𝑛𝜃 + 𝐵𝑐𝑜𝑠𝜃) = 0. 𝐴 = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼2 𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 𝐵 = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 − 𝑐𝑜𝑠𝛼1 𝑠𝑒𝑛𝛼2 𝑐𝑜𝑠𝜃 = 0. 𝐴𝑠𝑒𝑛𝜃 + 𝐵𝑐𝑜𝑠𝜃 = 0 𝜃 𝐵. 𝜃 = tan−1 (− 𝐴) + 𝑘𝜋 𝜃 𝜓. −𝜋, 𝜋. 39. −𝜋, 𝜋 𝑘 = 0,1. 𝑐𝑜𝑠𝜓.

(50) 𝑝. 𝜓 = tan−1 (− 𝑝12) + 𝑘𝜋. 𝑘 = 0,1. 11. 𝑝. 𝜓 = tan−1 (− 𝑝22) + 𝑘𝜋 21. 𝑘 = 0,1. 𝒄̂𝒊 𝛼𝑖. 𝑠𝑒𝑛𝜃 −𝐵 = 𝑐𝑜𝑠𝜃 𝐴. (67). 𝐴 = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼2 𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 𝐵 = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 − 𝑐𝑜𝑠𝛼1 𝑠𝑒𝑛𝛼2 𝛼𝑖 𝑠𝑒𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1. 𝐵2 𝑠𝑒𝑛 𝜃 = 2 𝑐𝑜𝑠 2 𝜃 = (1 − 𝑐𝑜𝑠 2 𝜃) 𝐴 2. 𝐵2 (1 + 2 ) 𝑐𝑜𝑠 2 𝜃 = 1 𝐴 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝜃 =. 𝑠𝑒𝑛𝜃 =. 𝐴 ±√𝐴2 + 𝐵 2. −𝐵 ±√𝐴2 + 𝐵 2 40. (68). (69).

(51) 𝒄̂𝟑 −𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃 −𝐴𝑐𝑜𝑠𝜙 𝐴𝑐𝑜𝑠𝛼3 1 −1 𝒄̂𝟑 = (−𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜃 ) = (−𝐴𝑠𝑒𝑛𝜙) = (𝐴𝑠𝑒𝑛𝛼3 ) ±√𝐴2 + 𝐵 2 ±√𝐴2 + 𝐵 2 𝐵 −𝐵 𝑠𝑒𝑛𝜃 𝒄̂𝟑 =. 𝐴𝑐𝑜𝑠𝛼3 (𝐴𝑠𝑒𝑛𝛼3 ) ±√𝐴2 + 𝐵 2 𝐵 −1. (70). 𝜙 = 𝛼3 𝒄̂𝟑 𝑝11 𝑝11 = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝑠𝑒𝑛𝜃 − 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝛼1 = 𝑝22 = 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝛼2 − 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝜃 =. −1 ±√𝐴2 + 𝐵 2 1 ±√𝐴2 + 𝐵 2. 𝑠𝑒𝑛𝜓 =. 𝑠𝑒𝑛2 𝜓 + 𝑐𝑜𝑠 2 𝜓 = 1. 𝑝22 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) (71) (𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ). −𝑝12 𝑐𝑜𝑠𝜓 𝑝11. (72). (73). 𝑐𝑜𝑠𝜓 𝑐𝑜𝑠𝜓 =. 𝑠𝑒𝑛𝜓 =. 𝑝11 ±√𝑝11 2 + 𝑝12 2 −𝑝12 ±√𝑝11 2 + 𝑝12 2 𝒄̂𝟏. 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 𝒄̂𝟏 = (−𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓) −𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓. 41. (74). (75).

(52) 𝑝11 𝑠𝑒𝑛𝜙. ±√𝑝11 2 + 𝑝12 2 ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ) −𝐵𝑝12 𝑠𝑒𝑛𝜙 𝑝11 𝑐𝑜𝑠𝜙 − ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ) ±√𝑝11 2 + 𝑝12 2 𝐴𝑝12. 𝒄̂𝟏 =. ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ). ( 𝒄̂𝟏 =. 𝐵𝑝12 𝑐𝑜𝑠𝜙. −. ). √𝐴2 + 𝐵 2 𝑝11 𝑠𝑒𝑛𝜙 − 𝐵𝑝12 𝑐𝑜𝑠𝜙. 1 ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ). (𝐵𝑝 𝑠𝑒𝑛𝜙 − √𝐴2 + 𝐵 2 𝑝 𝑐𝑜𝑠𝜙) 12 11 𝐴𝑝12 𝑝11. 𝜙 = 𝛼3. 𝑝12 = 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1. 𝛿 = ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ). −(𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 )𝑠𝑒𝑛𝛼3 − 𝐵𝑐𝑜𝑠 2 𝛼3 𝑠𝑒𝑛𝛼1 1 𝒄̂𝟏 = (𝐵𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 )𝑐𝑜𝑠𝛼3 ) 𝛿 𝐴𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1. 𝒄̂𝟏. (76). 𝛿 𝒄̂𝟐 𝐵𝑐𝑜𝑠𝜙𝑝11. 𝑝12 𝑠𝑒𝑛𝜙. ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ) ±√𝑝11 2 + 𝑝12 2 −𝑝12 𝑐𝑜𝑠𝜙 𝐵𝑠𝑒𝑛𝜙𝑝11 + ±√𝑝11 2 + 𝑝12 2 ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ) −𝐴𝑝11. −𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 𝒄̂𝟐 = ( 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 ) = −𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓. ±√(𝐴2 + 𝐵 2 )(𝑝11 2 + 𝑝12 2 ). (. 2 √ 2 1 𝐵𝑐𝑜𝑠𝜙𝑝11 + 𝐴 + 𝐵 𝑝12 𝑠𝑒𝑛𝜙 𝒄̂𝟐 = (−√𝐴2 + 𝐴2 𝑝 𝑐𝑜𝑠𝜙 + 𝐵𝑠𝑒𝑛𝜙𝑝 ) 12 11 𝛿 −𝐴𝑝11. 𝑝11 𝑝12. +. 𝜙 = 𝛼3. 42. ).

(53) −𝐵𝑐𝑜𝑠𝛼3 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) ±√𝐴2 1 𝒄̂𝟐 = 𝛿. +. 𝐵2. −√𝐴2 + 𝐵 2 𝑠𝑒𝑛𝛼1 𝑐𝑜𝑠 2 𝛼3 −. + √𝐴2 + 𝐵 2 𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3. 𝐵𝑠𝑒𝑛𝛼3 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ). ±√𝐴2 + 𝐵 2 𝐴(𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ). (. ±√𝐴2 + 𝐵 2. 1 ±√𝐴2 +𝐵2. −𝐵𝑐𝑜𝑠𝛼3 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) + (𝐴2 + 𝐵 2 )𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝒄̂𝟐 = ( −(𝐴2 + 𝐵 2 )𝑠𝑒𝑛𝛼1 𝑐𝑜𝑠 2 𝛼3 − 𝐵𝑠𝑒𝑛𝛼3 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) ) 𝛿√𝐴2 +𝐵2 𝐴(𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) 1. 2 1 −(𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 )𝑠𝑒𝑛𝛼3 − 𝐵𝑐𝑜𝑠 𝛼3 𝑠𝑒𝑛𝛼1 𝒄̂𝟏 = (𝐵𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 )𝑐𝑜𝑠𝛼3 ) 𝛿 𝐴𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1. −𝐵𝑐𝑜𝑠𝛼3 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) + (𝐴2 + 𝐵2 )𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 𝒄̂𝟐 = ( −(𝐴2 + 𝐵2 )𝑠𝑒𝑛𝛼1 𝑐𝑜𝑠 2 𝛼3 − 𝐵𝑠𝑒𝑛𝛼3 (𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) ) 2 2 𝛿√𝐴 + 𝐵 𝐴(𝐵𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼3 + 𝐴𝑐𝑜𝑠𝛼1 ) 1. 𝐴𝑐𝑜𝑠𝛼3 𝒄̂𝟑 = (𝐴𝑠𝑒𝑛𝛼3 ) ±√𝐴2 + 𝐵2 𝐵 −1. 43. ).

(54) 𝑠𝑒𝑛𝜓 =. −𝑝22 𝑐𝑜𝑠𝜓 𝑝21. (79) 𝑠𝑒𝑛𝜓. 𝑝12 → 𝑝22. 𝑐𝑜𝑠𝜓 =. 𝑐𝑜𝑠𝜓. 𝑝11 → 𝑝21. 𝑝21. 𝑠𝑒𝑛𝜓 =. ±√𝑝21 2 + 𝑝22 2. −𝑝22 ±√𝑝21 2 + 𝑝22 2. 𝒄̂𝟏. 𝑠𝑒𝑛𝜙𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 𝒄̂𝟏 = (−𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜓 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓) −𝑐𝑜𝑠𝜃𝑠𝑒𝑛𝜓 𝑝21 𝑠𝑒𝑛𝜙. ±√𝑝21 2 + 𝑝22 2 ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ) 𝐵𝑝22 𝑠𝑒𝑛𝜙 𝑝21 𝑐𝑜𝑠𝜙 − − ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ) ±√𝑝21 2 + 𝑝22 2 𝐴𝑝22. 𝒄̂𝟏 =. ( 𝒄̂𝟏 =. 𝐵𝑝22 𝑐𝑜𝑠𝜙. −. ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ) 1. ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ). ). −√𝐴2 + 𝐵 2 𝑝21 𝑠𝑒𝑛𝜙 − 𝐵𝑝22 𝑐𝑜𝑠𝜙 ( −𝐵𝑝 𝑠𝑒𝑛𝜙 + √𝐴2 + 𝐵 2 𝑝 𝑐𝑜𝑠𝜙 ) 22. 21. 𝐴𝑝22. 𝑝21 = −𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑝22 𝜌 = ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ). 44. 𝜙 = 𝛼3.

(55) −√𝐴2 + 𝐵 2 𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑠𝑒𝑛𝛼3 − 𝒄̂𝟏 =. 𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑐𝑜𝑠𝛼3 ±√𝐴2 + 𝐵 2. −𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑠𝑒𝑛𝛼3. 1 𝜌. + √𝐴2 + 𝐵 2 𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ±√𝐴2 + 𝐵 2 𝐴(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ). (. ). ±√𝐴2 + 𝐵 2. 1. ±√𝐴2 + 𝐵 2. ±√𝐴2 +𝐵2. 𝜌. −(𝐴2 + 𝐵 2 )𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑠𝑒𝑛𝛼3 − 𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑐𝑜𝑠𝛼3 𝒄̂𝟏 = ( −𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑠𝑒𝑛𝛼3 + (𝐴2 + 𝐵 2 )𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ) 𝜌√𝐴2 +𝐵2 𝐴(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ) 1. 𝒄̂𝟐 𝐵𝑐𝑜𝑠𝜙𝑝21. +. 𝑝22 𝑠𝑒𝑛𝜙. ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ) ±√𝑝21 2 + 𝑝22 2 −𝑝22 𝑐𝑜𝑠𝜙 𝐵𝑠𝑒𝑛𝜙𝑝21 + ±√𝑝21 2 + 𝑝22 2 ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ) −𝐴𝑝21. −𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜓 𝒄̂𝟐 = ( 𝑐𝑜𝑠𝜙𝑠𝑒𝑛𝜓 − 𝑠𝑒𝑛𝜙𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜓 ) = −𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜓. ±√(𝐴2 + 𝐵 2 )(𝑝21 2 + 𝑝22 2 ). (. 2 √ 2 1 𝐵𝑐𝑜𝑠𝜙𝑝21 + 𝐴 + 𝐵 𝑝22 𝑠𝑒𝑛𝜙 𝒄̂𝟐 = (−√𝐴2 + 𝐵 2 𝑝 𝑐𝑜𝑠𝜙 + 𝐵𝑠𝑒𝑛𝜙𝑝 ) 22 21 𝜌 −𝐴𝑝21. 𝑝21 = −𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑝22. 1 𝒄̂𝟐 = ( 𝜌. 𝜙 = 𝛼3. −𝐵𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 + (𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑠𝑒𝑛𝛼3 −(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑐𝑜𝑠𝛼3 − 𝐵𝑠𝑒𝑛2 𝛼3 𝑐𝑜𝑠𝛼2 ) 𝐴𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2. 45. ).

(56) (𝐴2 + 𝐵2 )𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑠𝑒𝑛𝛼3 + 𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑐𝑜𝑠𝛼3 𝒄̂𝟏 = ( 𝐵(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑠𝑒𝑛𝛼3 − (𝐴2 + 𝐵2 )𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ) 𝜌√𝐴2 + 𝐵2 −𝐴(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 ) 1. −𝐵𝑐𝑜𝑠𝛼3 𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 + (𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑠𝑒𝑛𝛼3 1 𝒄̂𝟐 = ( −(𝐴𝑠𝑖𝑛𝛼2 + 𝐵𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 )𝑐𝑜𝑠𝛼3 − 𝐵𝑠𝑒𝑛2 𝛼3 𝑐𝑜𝑠𝛼2 ) 𝜌 𝐴𝑠𝑒𝑛𝛼3 𝑐𝑜𝑠𝛼2 𝒄̂𝟑 =. 𝐴𝑐𝑜𝑠𝛼3 (𝐴𝑠𝑒𝑛𝛼3 ) ±√𝐴2 + 𝐵2 𝐵 1. 46.

(57) α1 = α2 = α3 =. 47. π 2.

(58) 48.

(59) 𝐴=0 𝑓(𝛼1 , 𝛼2 , 𝛼3 ) = 𝑠𝑒𝑛𝛼1 𝑠𝑒𝑛𝛼2 𝑠𝑒𝑛𝛼3 + 𝑐𝑜𝑠𝛼1 𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛼3 = 0. 𝑠𝑒𝑛𝛼2 = 0 𝑦 𝑐𝑜𝑠𝛼3 = 0 𝑠𝑒𝑛𝛼3 = 0 𝑦 𝑐𝑜𝑠𝛼1 = 0 𝑠𝑒𝑛𝛼1 = 0 𝑦 𝑐𝑜𝑠𝛼2 = 0. 𝑝12 = 0. 𝑝22 = 0. 𝑠𝑒𝑛𝜓 = 0 𝛼3 = 𝜋. 𝛼3 = − 2. 0 𝑹𝟏𝒂 = ( 𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃. −1 0 0 𝑠𝑒𝑛𝜃) 0 𝑐𝑜𝑠𝜃. 0 𝑹𝟐𝒂 = ( −𝑐𝑜𝑠𝜃 −𝑠𝑒𝑛𝜃. 1 0 0 −𝑠𝑒𝑛𝜃) 0 𝑐𝑜𝑠𝜃 𝑝11 = 0. 𝑝21 = 0. 𝑐𝑜𝑠𝜓 = 0. 𝑹𝟏𝒃. 𝑐𝑜𝑠𝜃 =( 0 𝑠𝑒𝑛𝜃 49. 𝑠𝑒𝑛𝜃 0 𝑐𝑜𝑠𝜃. 0 −1) 0. 𝜋 2.

(60) 𝑹𝟐𝒃 = (. 𝑝11 = 0. 𝑐𝑜𝑠𝜃 0 −𝑠𝑒𝑛𝜃. −𝑠𝑒𝑛𝜃 0 −𝑐𝑜𝑠𝜃. 0 1) 0. 𝑐𝑜𝑠𝜃 = 0. 0 −𝑠𝑒𝑛(𝜙 − 𝜓) 𝑐𝑜𝑠(𝜙 − 𝜓) 𝑹𝟏𝒄 = ( 0 𝑐𝑜𝑠(𝜙 − 𝜓) 𝑠𝑒𝑛(𝜙 − 𝜓)) −1 0 0 0 𝑹𝟐𝒄 = (0 1. −𝑠𝑒𝑛(𝜙 − 𝜓) 𝑐𝑜𝑠(𝜙 − 𝜓) 0. −𝑐𝑜𝑠(𝜙 − 𝜓) −𝑠𝑒𝑛(𝜙 − 𝜓)) 0. 𝑐𝑜𝑠𝜃 = 0. 𝑐𝑜𝑠𝜃 = 0. 50.

(61) 51.

(62) 52.

(63) 53.

(64) 54.

(65) .. 55.

(66) 56.

(67) .. 57.

(68) 𝑍0. 𝑍𝑖−1 𝑍𝑛−1. 58.

(69)