Construcción y pruebas de un módulo didáctico de un manipulador paralelo con actuadores lineales
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(2)
(3)
(4)
(5) 5.
(6) 6.
(7) 7.
(8) 8.
(9) 9. PRPaR PPa4S. 1P1R1S 3PPa4S. PPa4S.
(10) 10. 𝑄𝑖. 𝐻𝑖.
(11) 11.
(12) 12.
(13) 13.
(14) 14.
(15) 15.
(16) 16. • • •.
(17) 17.
(18) 18.
(19) 19.
(20) 20.
(21) 21.
(22) 22.
(23) 23.
(24) 24.
(25) 25.
(26) 26.
(27) 27.
(28) 28.
(29) 29.
(30) 30.
(31) 31. •. •. •. •. •. •.
(32) 32. •. •.
(33) 33.
(34) 34.
(35) 35.
(36) 36.
(37) 37.
(38) 38. • • • • •. • •. •. •.
(39) 39. 𝑔. 𝑀 = 𝑑 (𝑛 − 𝑔 − 1) + ∑ 𝑓𝑖 − 𝑓𝑔 𝑖=1. 𝑑. 𝑓𝑖 𝑓𝑔.
(40) 40. • • • • •. 𝑑=6 𝐿=3 𝑓𝑙𝑗. 𝑛 = 2 + 𝑛𝑐𝑐 𝐿. ∑(𝑓𝑙𝑗 − 1) Binarios 𝑛𝑐𝑐. 𝑗=1 𝐿. ∑(𝑓𝑙𝑗 − 2) Ternarios {𝑗=1 𝑛𝑐𝑐.
(41) 41. 𝐿. 𝑔 = ∑ 𝑓𝑙𝑗 𝑗=1. 𝐿. 𝑔. 𝐿. 𝑀 = 6 [2 + ∑(𝑓𝑙𝑗 − 1) − ∑(𝑓𝑙𝑗 − 1)] + ∑ 𝑓𝑙𝑗 − 𝑓𝑔 𝑗=1. 𝑗=1. 𝑗=1 𝑔. 𝑀 = 6[1 − 𝐿] + ∑ 𝑓𝑙𝑗 − 𝑓𝑔 𝑗=1. 𝑓𝑔 = 0. 𝑀=3. 𝐿=3 𝑔. ∑ 𝑓𝑗 = 15 𝑗=1.
(42) 42. 𝑀=3 𝐿=3 𝑓𝑔 = 0. 𝑓𝑗 = 5.
(43) 43. 𝑓𝑗 = 5 1P1R1S 1P2R1H 1P1P1R1H 1P2P1H 1P4R 1P1P3R 1P2P2R 1P3P1R 1P4P. 𝐿. 𝑔. 𝐿. 𝑀 = 6 [2 + ∑(𝑓𝑙𝑗 − 2) − ∑(𝑓𝑙𝑗 − 1)] + ∑ 𝑓𝑗 − 𝑓𝑔 𝑗=1. 𝑗=1. 𝑗=1 𝑔. 𝑀 = 6[1 − 2𝐿] + ∑ 𝑓𝑗 − 𝑓𝑔 𝑗=1. 𝑓𝑔 = 0. 𝑀=3. 𝑔. ∑ 𝑓𝑗 − 𝑓𝑔 = 33 𝑗=1. 𝐿=3.
(44) 44. 𝑁. ∑ 𝑓𝐿𝑖 − ̅̅̅̅ 𝑓𝑔𝐿 = 11 𝑖=1. 𝑓𝐿𝑖 𝑓𝑔𝐿. PRPaR.
(45) 45. PPa4S. 𝑃𝑎4𝑆 PRPaR • • •. 1.
(46) 46. •. PPa4s • • •. 𝑓𝑔𝐿 = −4. 𝑁. ∑ 𝑓𝐿𝑖 = 7 𝑖=1.
(47) 47. 𝑓𝑔𝐿 = 2. 𝑁. ∑ 𝑓𝐿𝑖 = 13 𝑖=1. 1P1R1S 1P2R1H 1P1P1R1H 1P2P1H 1P4R 1P1P3R 1P2P2R 1P3P1R 1P4P 1PRPAR 1PPA4S.
(48) 48. • • • •.
(49) 49. •. •.
(50) 50. ≥. • (1P1R1S). (1P1R1S) 1P1R1S. 𝑓𝑔 = 0. (1PPA4S ).
(51) 51. 𝑓𝑔 = 0 𝑓𝑔 = −3. ∑ 𝑓𝑖 = 2. 3PPA4S. 3PPa4S.
(52) 52. ←LA ─LB ─LC.
(53) 53.
(54) 54. φ. φ. φ.
(55) 55. PPa4S. β. β. β.
(56) 56. 𝑝. 𝑘2. 𝑀𝑀𝑃 = ∑ 𝑓𝑖 − ∑ 𝑆𝐿𝑗 + 𝑆𝑛𝑀𝑃 ⁄ − 𝑟1 𝑖=1. 𝑗=1. 1. 𝑝 𝑘2 𝑟1 𝑛. 𝑝. ∑ 𝑓𝑖 𝑖=1. 𝐿𝑗. 𝑆𝐿𝑗 𝑘2. ∑ 𝑆𝐿𝑗 𝑗=1. 𝑆𝑛𝑀𝑃 ⁄ 1. 𝑛. 𝑆𝑛𝑀𝑃 ⁄ = ⋂ 𝐿𝑖 = 𝐿𝐴 ∩ 𝐿𝐵 ∩ … .∩ 𝐿𝑛 1. 𝑖=1.
(57) 57. 𝑃. 𝑇𝑀𝑃 = ∑ 𝑓𝑖 − 𝑝 − 𝑚 𝑖=1. 𝑚. 𝑘2. 𝑁𝑀𝑃 = 6𝑞 − ∑ 𝑆𝐿𝑖 + 𝑆𝑛𝑀𝑃 ⁄ − 𝑟1 𝑖=1. 𝑞 𝑞 = 𝑝−𝑚+1 𝑘2 𝑟1. 1.
(58) 58. 𝐿𝐴 (1𝐴 ≡ 0−2𝐴 − 3𝐴 − 4𝐴 − 5𝐴 ≡ 5) 𝐿𝐵 (1𝐵 ≡ 0−2𝐵 − 3𝐵 − 4𝐵 − 5𝐵 ≡ 5) 𝐿𝐶 (1𝐶 ≡ 0−2𝐶 − 3𝐶 − 4𝐶 − 5𝐶 ≡ 5) ≡ 1𝐴 ≡ 0 −. • • • • •. 𝑚 = 11−→ 𝑝 = 15−→ 𝑞 =𝑝−𝑚+1=5→ 𝑘2 = 3−→ 5 ≡ 5𝐴 ≡ 5𝐵 ≡ 5𝑐 −→. •. 1 ≡ 1𝐴 ≡ 1𝐵 ≡ 1𝑐 ≡ 0−→ PPa4S. 𝜔𝑖 𝜔𝑖+1 𝐿𝐵 𝑖 = 5. 𝜐𝑥 , 𝜐𝑦 , 𝜐𝑧 𝐿𝐴 𝑖 = 3. 𝑖=1 𝐿𝐶 𝑅𝐿𝐴 = (𝜐𝑥 , 𝜐𝑦 , 𝜐𝑧 , 𝜔1 , 𝜔2 ) 𝑅𝐿𝐵 = (𝜐𝑥 , 𝜐𝑦 , 𝜐𝑧 , 𝜔3 , 𝜔4 ) 𝑅𝐿𝐶 = (𝜐𝑥 , 𝜐𝑦 , 𝜐𝑧 , 𝜔5 , 𝜔6 ). 𝑆𝐸𝑖 = 5 = dim(𝐿𝑖 ),. 𝑖 = 𝐴, 𝐵, 𝐶.
(59) 59.
(60) 60. 𝑛. 𝑆5𝐷⁄ 1. = ⋂ 𝐿𝑖 = 𝐿𝐴 ∩ 𝐿𝐵 ∩ 𝐿𝐶 = (𝜐𝑥 , 𝜐𝑦 , 𝜐𝑧 ) 𝑖=1. 𝑃. 𝑘2. 𝑀𝑀𝑃 = ∑ 𝑓𝑖 − ∑ 𝑆𝐿𝑗 + 𝑆𝑛𝑀𝑃 ⁄ − 𝑟1 𝑖=1. 𝑗=1. 1.
(61) 61. 𝑃. ∑ 𝑓𝑖 = Dim(𝑢) = 3(12) + 3 𝑖=1 𝑃. ∑ 𝑓𝑖 = 39 𝑖=1. (Pa4S ).
(62) 62. 𝑟1 = 𝑘2 (6) = 18. 3. 𝑟𝑀𝑃 = ∑ 𝑆𝐸𝑗 − 𝑆5𝑀𝑃 + 𝑟1 ⁄ 𝑗=1. 1. 𝑟𝐷 = 15 − 3 + 18 = 30. 15. 3. 𝑀𝑀𝑃 = ∑ 𝑓𝑖 − ∑ 𝑆𝐸𝑗 + 𝑆5𝑀𝑃 − 𝑟1 ⁄ 𝑖=1. 𝑗=1. 1. 𝑀𝑀𝑃 = 39 − 15 + 3 − 18 = 9. 𝑁𝑀𝑃 = 30 − 15 + 3 − 18 = 0. 𝑇𝑀𝑃 = 39 − 15 − 18 = 6.
(63) 63. (Pa4S ).
(64) 64. 𝑀𝑀𝑃𝑇𝑜𝑡𝑎𝑙 = 𝑀𝑀𝑃 − 𝑇𝑀𝑃 𝑀𝑀𝑃𝑇𝑜𝑡𝑎𝑙 = 9 − 6 = 3. 𝑞𝐴 , 𝑞𝐵 , 𝑞𝐶.
(65) 65.
(66) 66. 𝐴𝐴1 = 𝑎𝐴𝑥 𝒊 + 𝑎𝐴𝑦 𝒋 𝐴𝐵1 = 𝑎𝐵𝑥 𝒊 + 𝑎𝐵𝑦 𝒋 𝐴𝐶1 = 𝑎𝐶𝑥 𝒊 + 𝑎𝐶𝑦 𝒋. 𝑂. 𝑇𝐴1𝑖. cos(𝜑𝑖 − 𝜋) −𝑠in(𝜑𝑖 − 𝜋) =[ 0 0 −cos(𝜑𝑖 ) sin(𝜑𝑖 ) =[ 0 0. sin(𝜑𝑖 − 𝜋) cos(𝜑𝑖 − 𝜋) 0 0 −sin(𝜑𝑖 ) −cos(𝜑𝑖 ) 0 0. 0 𝑎𝑖𝑥 0 𝑎𝑖𝑦 ]= 1 0 0 1. 0 𝑎𝑖𝑥 0 𝑎𝑖𝑦 ] 1 0 0 1.
(67) 67. 1 0 𝐴1𝑖 𝑇𝑄𝑖 = [ 0 0. 0 1 0 0. 0 0 0 0 ] 1 𝑞𝑖 0 1.
(68) 68. 𝑄’𝐴 = 𝑎𝐴𝑥 𝒊 + 𝑎𝐴𝑦 𝒋 + 𝑞𝐴 𝒌 𝑄’𝐵 = 𝑎𝐵𝑥 𝒊 + 𝑎𝐵𝑦 𝒋 + 𝑞𝐵 𝒌 𝑄’𝐶 = 𝑎𝐶𝑥 𝒊 + 𝑎𝐶𝑦 𝒋 + 𝑞𝐶 𝒌 πa, πb y πc. πi. πA, πB y πC γi ρi.
(69) 69. 𝑄𝑖. 𝑇𝑄´𝑖. cos(𝜌𝑖 )cos(𝛾𝑖 ) −sin(𝜌𝑖 ) sin(𝜌𝑖 )cos(𝛾𝑖 ) cos(ρ𝑖 ) =[ −𝑠𝑖𝑛𝑛(𝛾𝑖 ) 0 0 0. 𝑃 = 𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌. cos(𝜌𝑖 )sin(𝛾𝑖 ) sin(𝜌𝑖 )cos(𝛾𝑖 ) cos(𝛾𝑖 ) 0. 0 0 ] 0 1.
(70) 70. 𝐻𝐴 = 𝑏𝐴𝑢 𝒖 + 𝑏𝐴𝑣 𝒗 𝐻𝐵 = 𝑏𝐵𝑢 𝒖 + 𝑏𝐵𝑣 𝒗 𝐻𝐶 = 𝑏𝐶𝑢 𝒖 + 𝑏𝐶𝑣 𝒗.
(71) 71.
(72) 72. 𝑃0 = 𝑧0 𝒌.
(73) 73.
(74) 74. 𝜆𝑖 = |𝑓̂ 𝑖 ∙ 𝑣̂𝑖 | = |cos 𝜓| = |sin 𝛾 |. 𝜆𝑖 𝑣𝑖 |𝑣𝑖 | 𝑓𝑖 𝑓̂ 𝑖 = | 𝑓𝑖 |. 0 ≤ 𝜆𝑖 ≤ 1. 𝑣̂𝑖 =. 𝜓 𝛾 𝛾 + 𝜓 = 90𝑜.
(75) 75. 𝛾. 𝜓. 𝛾. 𝜓. sin 45𝑜 ≈ 0,7.
(76) 76. 𝑅 𝑟 𝑧máx 𝑧mín 𝐿.
(77) 77.
(78) 78.
(79) 79. sin 𝜓máx =. 𝑅 + (𝑟0 − 𝑟) 𝐿. sin 𝜓mín =. 𝑅 − (𝑟0 + 𝑟) 𝐿. 𝜌 = 𝜓máx − 𝜓mín. 𝐿=. 2𝑟0 sin 𝜓máx − sin 𝜓mín. 𝑅 − 𝑟 = 𝐿 ∙ 𝑠in 𝜓mín + 𝑟0. 𝑧mín − 𝑧0 = 𝐿 ⋅ cos 𝜓máx 𝑧máx − 𝑧0 = 𝐿 ⋅ cos 𝜓mín + ℎ. • •. 𝑟0 = 75 mm ℎ = 200 mm.
(80) 80. • • • • • •. 𝜌 = 22,5𝑜 𝛾mín = 45𝑜 𝜓máx = 45𝑜 𝛾máx = 77,5𝑜 𝜓mín = 22,5𝑜. 𝜓máx. 𝜓mín. 𝜓máx. 𝜓mín.
(81) 81.
(82) 82. 𝐿 (𝑎𝑖𝑥 , 𝑎𝑖𝑦 , 𝑞𝑖𝑗 ). 𝑞𝑖𝑗 𝑞𝑖𝑗 ∈ [𝑧𝑖 mín , 𝑧𝑖 máx ]. (𝑥𝑖 , 𝑦𝑖 , 𝑧mín ). (𝑥𝑖 , 𝑦𝑖 , 𝑧máx). |𝑧𝑖 𝑚ín − 𝑧𝑖 𝑚áx| ≥ 2𝐿 |𝑧𝑖 𝑚ín − 𝑧𝑖 𝑚áx| < 2𝐿 |𝑧𝑖 mín − 𝑧𝑖 máx | ≥ 2𝐿. 𝐿 𝐿. |𝑧𝑖 mín − 𝑧𝑖 máx|.
(83) 83. |𝑧𝑖 mín − 𝑧𝑖 máx | < 2𝐿 (𝑥 − 𝑥𝑖 )2 + (𝑦 − 𝑦𝑖 )2 + (𝑧 − 𝑧𝑖 máx)2 = 𝐿2. (𝑥 − 𝑥𝑖 )2 + (𝑦 − 𝑦𝑖 )2 + (𝑧 − 𝑧𝑖 mín )2 = 𝐿2. |𝑧𝑖 𝑚ín − 𝑧𝑖 𝑚áx | = 310,2531 mm < 2𝐿 = 920 mm.
(84) 84.
(85) 85.
(86) 86.
(87) 87.
(88) 88. 15363236,80. 11122421,47. 3185626,96.
(89) 89.
(90) 90. SA , SB 𝐶𝑆𝐴 , 𝐶𝑆𝐵. 𝑄𝐴 , 𝑄𝐵. 𝑄𝐶. 𝐶𝑆𝐶. SC.
(91) 91. 𝑅 𝑟 𝐿. SA 𝐶𝑆𝐵. 𝑥𝑆𝐵 , 𝑦𝑆𝐵 , 𝑧𝑆𝐵. 𝐶𝑆𝐴. SC. 𝐶𝑆𝐶. 𝑥𝑆𝐴 , 𝑦𝑆𝐴, 𝑧𝑆𝐴. 𝑥𝑆𝐶 , 𝑦𝑆𝐶 , 𝑧𝑆𝐶 𝑅−𝑟. 𝑞𝐴 , 𝑞𝐵 , 𝑞𝐶. 𝑃⃗. SB.
(92) 92. 𝐶𝑆𝑖. 𝑥𝑆𝑖 , 𝑦𝑆𝑖 , 𝑧𝑆𝑖. 𝑧𝑆𝑖 = 𝑞𝑖. 𝑧𝑆𝑖 𝑞𝑖.
(93) 93. 𝐶𝑆𝐴 𝐶𝑆𝐵 𝐶𝑆𝐶. (𝑥𝑆𝐴 , 𝑦𝑆𝐴, 𝑧𝑆𝐴) = ((𝑅 − 𝑟)cos𝜑𝐴 , (𝑅 − 𝑟)sin𝜑𝐴, 𝑧𝑆𝐴) (𝑥𝑆𝐵 , 𝑦𝑆𝐵 , 𝑧𝑆𝐵 ) = ((𝑅 − 𝑟)cos𝜑𝐵 , (𝑅 − 𝑟)sin𝜑𝐵 , 𝑧𝑆𝐵 ) (𝑥𝑆𝐶 , 𝑦𝑆𝐶 , 𝑧𝑆𝐶 ) = ((𝑅 − 𝑟)𝑐𝑜𝑠𝜑𝐶 , (𝑅 − 𝑟)sin𝜑𝐶 , 𝑧𝑆𝐶 ) φ. φ. φ. 𝐿𝑅𝑟 = 𝑅 − 𝑟. 𝐶𝑆𝐴 𝐶𝑆𝐵 𝐶𝑆𝐶. (𝑥𝑆𝐴 , 𝑦𝑆𝐴, 𝑧𝑆𝐴) = (0 , 𝐿𝑅𝑟 , 𝑞𝐴 ) 3. 1. (𝑥𝑆𝐵 , 𝑦𝑆𝐵 , 𝑧𝑆𝐵 ) = (√ ∙ 𝐿𝑅𝑟 , − ∙ 𝐿𝑅𝑟 , 𝑞𝐵 ) 2. (𝑥𝑆𝐶 , 𝑦𝑆𝐶 , 𝑧𝑆𝐶 ) =. √3 (− 2. 2. 1. ∙ 𝐿𝑅𝑟 , − ∙ 𝐿𝑅𝑟 , 𝑞𝐶 ) 2. 𝑆𝑖. 𝑥𝑆𝑖 , 𝑦𝑆𝑖 , 𝑧𝑆𝑖. 𝑆𝑖 : (𝑥 − 𝑥𝑆𝑖 )2 + (𝑦 − 𝑦𝑆𝑖 )2 + (𝑧 − 𝑧𝑆𝑖 )2 = 𝐿2 𝑆𝐴 , 𝑆𝐵 𝑦 𝑆𝐶. 𝑆𝐴 : 𝑥 2 + (𝑦 − 𝐿𝑅𝑟 )2 + (𝑧 − 𝑧𝑆𝐴 )2 = 𝐿2 2. 2 1 √3 (𝑦 ) 𝑆𝐵 : (𝑥 − ∙ 𝐿𝑅𝑟 ) + + ∙ 𝐿𝑅𝑟 + (𝑧 − 𝑧𝑆𝐵 )2 = 𝐿2 2 2 2. 2 1 √3 𝑆𝐶 : (𝑥 + ∙ 𝐿𝑅𝑟 ) + (𝑦 + ∙ 𝐿𝑅𝑟 ) + (𝑧 − 𝑧𝑆𝐶 )2 = 𝐿2 2 2. 𝑧𝑆𝐴, 𝑧𝑆𝐵 , 𝑧𝑆𝐶. 𝑞𝐴 𝑞𝐵 𝑞𝐶.
(94) 94. 𝑆𝐴 : 𝑢2 + (𝑣 − 𝐿𝑅𝑟 )2 + (𝑤 − 𝑞𝐴 )2 = 𝐿2 2. 2 1 √3 𝑆𝐵 : (𝑢 − ∙ 𝐿𝑅𝑟 ) + (𝑣 + ∙ 𝐿𝑅𝑟 ) + (𝑤 − 𝑞𝐵 )2 = 𝐿2 2 2 2. 2 1 √3 𝑆𝐶 : (𝑢 + ∙ 𝐿𝑅𝑟 ) + (𝑣 + ∙ 𝐿𝑅𝑟 ) + (𝑤 − 𝑞𝐶 )2 = 𝐿2 2 2. 𝑞𝐴 𝑞𝐵. 𝑞𝐶. 𝑞𝐴 = 𝑤 + √𝐿2 − 𝑢2 − (𝑣 − 𝐿𝑅𝑟 )2 2. 2 1 √3 2 √ 𝑞𝐵 = 𝑤 + 𝐿 − (𝑢 − ∙ 𝐿𝑅𝑟 ) − (𝑣 + ∙ 𝐿𝑅𝑟 ) 2 2 2. 𝑞𝐶 = 𝑤 +. √𝐿2. 2 1 √3 − (𝑢 + ∙ 𝐿𝑅𝑟 ) − (𝑣 + ∙ 𝐿𝑅𝑟 ) 2 2. 𝑞𝐴 𝑞𝐵 𝑞𝐶 𝑞𝐴 𝑞𝐵. 𝑞𝐶. (𝑢 − 𝑥𝑆𝐴 )2 + (𝑣 − 𝑦𝑆𝐴)2 + (𝑤 − 𝑞𝐴 )2 = 𝐿2 (𝑢 − 𝑥𝑆𝐵 )2 + (𝑣 − 𝑦𝑆𝐵 )2 + (𝑤 − 𝑞𝐵 )2 = 𝐿2 (𝑢 − 𝑥𝑆𝐶 )2 + (𝑣 − 𝑦𝑆𝐶 )2 + (𝑤 − 𝑞𝐶 )2 = 𝐿2. (𝑥𝑆𝐴 , 𝑦𝑆𝐴) = (0 , 𝐿𝑅𝑟 ). (𝑥𝑆𝐵 , 𝑦𝑆𝐵 ) = (. 1 √3 𝐿𝑅𝑟 , − 𝐿𝑅𝑟 ) 2 2. (𝑥𝑆𝐶 , 𝑦𝑆𝐶 ) = (−. 1 √3 𝐿𝑅𝑟 , − 𝐿𝑅𝑟 ) 2 2.
(95) 95. 𝑢2 + 𝑣 2 + 𝑤 2 + 𝑥𝑆𝐴 2 + 𝑦𝑆𝐴2 + 𝑞𝐴 2 − 2𝑢𝑥𝑆𝐴 − 2𝑣𝑦𝑆𝐴 − 2𝑤𝑞𝐴 = 𝐿2 𝑢2 + 𝑣 2 + 𝑤 2 + 𝑥𝑆𝐵 2 + 𝑦𝑆𝐵 2 + 𝑞𝐵 2 − 2𝑢𝑥𝑆𝐵 − 2𝑣𝑦𝑆𝐵 − 2𝑤𝑞𝐵 = 𝐿2 𝑢2 + 𝑣 2 + 𝑤 2 + 𝑥𝑆𝐶 2 + 𝑦𝑆𝐶 2 + 𝑞𝐶 2 − 2𝑢𝑥𝑆𝐶 − 2𝑣𝑦𝑆𝐶 − 2𝑤𝑞𝐶 = 𝐿2. 𝑟𝑆𝐴 = 𝑥𝑆𝐴 2 + 𝑦𝑆𝐴 2 + 𝑞𝐴 2. 𝑟𝑆𝐵 = 𝑥𝑆𝐵 2 + 𝑦𝑆𝐵 2 + 𝑞𝐵 2. 𝑟𝑆𝐶 = 𝑥𝑆𝐶 2 + 𝑦𝑆𝐶 2 + 𝑞𝐶 2. 𝑢2 + 𝑣 2 + 𝑤 2 + 𝑟𝑆𝐴 − 2𝑢𝑥𝑆𝐴 − 2𝑣𝑦𝑆𝐴 − 2𝑤𝑞𝐴 = 𝐿2 𝑢2 + 𝑣 2 + 𝑤 2 + 𝑟𝑆𝐵 − 2𝑢𝑥𝑆𝐵 − 2𝑣𝑦𝑆𝐵 − 2𝑤𝑞𝐵 = 𝐿2 𝑢2 + 𝑣 2 + 𝑤 2 + 𝑟𝑆𝐶 − 2𝑢𝑥𝑆𝐶 − 2𝑣𝑦𝑆𝐶 − 2𝑤𝑞𝐶 = 𝐿2. 𝑟𝑆𝐴 − 𝑟𝑆𝐵 − 2𝑢(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) − 2𝑣(𝑦𝑆𝐴 − 𝑦𝑆𝐵 ) − 2𝑤(𝑞𝐴 − 𝑞𝐵 ) = 0 𝑟𝑆𝐴 − 𝑟𝑆𝐶 − 2𝑢 (𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − 2𝑣(𝑦𝑆𝐴 − 𝑦𝑆𝐶 ) − 2𝑤(𝑞𝐴 − 𝑞𝐶 ) = 0. 𝑟𝑆𝐴 − 𝑟𝑆𝐵 − 𝑤(𝑞𝐴 − 𝑞𝐵 ) 2 𝑟𝑆𝐴 − 𝑟𝑆𝐶 𝑢(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) + 𝑣 (𝑦𝑆𝐴 − 𝑦𝑆𝐶 ) = − 𝑤(𝑞𝐴 − 𝑞𝐶 ) 2. 𝑢(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) + 𝑣(𝑦𝑆𝐴 − 𝑦𝑆𝐵 ) =. (𝑟𝑆𝐴 − 𝑟𝑆𝐶 )(𝑦𝑆𝐴 − 𝑦𝑆𝐵 ) − (𝑟𝑆𝐴 − 𝑟𝑆𝐵 )(𝑦𝑆𝐴 − 𝑦𝑆𝐶 ) ] 𝑢=[ 2(𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − 2(𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) (𝑞𝐴 − 𝑞𝐶 )(𝑦𝑆𝐴 − 𝑦𝑆𝐵 ) − (𝑞𝐴 − 𝑞𝐵 )(𝑦𝑆𝐴 − 𝑦𝑆𝐶 ) ] +𝑤[ (𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) − (𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ).
(96) 96. (𝑟𝑆𝐴 − 𝑟𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − (𝑟𝑆𝐴 − 𝑟𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) ] 𝑣=[ 2(𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − 2(𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) (𝑞𝐴 − 𝑞𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) − (𝑞𝐴 − 𝑞𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) ] +𝑤[ (𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − (𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ). 𝑎=. (𝑟𝑆𝐴 − 𝑟𝑆𝐶 )(𝑦𝑆𝐴 − 𝑦𝑆𝐵 ) − (𝑟𝑆𝐴 − 𝑟𝑆𝐵 )(𝑦𝑆𝐴 − 𝑦𝑆𝐶 ) 2(𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − 2(𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ). 𝑏= 𝑐=. (𝑞𝐴 − 𝑞𝐶 )(𝑦𝑆𝐴 − 𝑦𝑆𝐵 ) − (𝑞𝐴 − 𝑞𝐵 )(𝑦𝑆𝐴 − 𝑦𝑆𝐶 ) (𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) − (𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ). (𝑟𝑆𝐴 − 𝑟𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − (𝑟𝑆𝐴 − 𝑟𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) 2(𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − 2(𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ). 𝑑=. (𝑞𝐴 − 𝑞𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ) − (𝑞𝐴 − 𝑞𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) (𝑦𝑆𝐴 − 𝑦𝑆𝐵 )(𝑥𝑆𝐴 − 𝑥𝑆𝐶 ) − (𝑦𝑆𝐴 − 𝑦𝑆𝐶 )(𝑥𝑆𝐴 − 𝑥𝑆𝐵 ). 𝑢 = 𝑎 + 𝑤𝑏 𝑣 = 𝑐 + 𝑤𝑑. (𝑎 + 𝑤𝑏 − 𝑥𝑆𝐴 )2 + (𝑐 + 𝑤𝑑 − 𝑦𝑆𝐴)2 + (𝑤 − 𝑞𝐴 )2 = 𝐿2. 2𝑏(𝑎 − 𝑥𝑆𝐴 ) + 2𝑑 (𝑐 − 𝑦𝑆𝐴 ) − 2𝑞𝐴 ]𝑤 𝑤2 + [ 𝑏2 + 𝑑2 + 1 (𝑎 − 𝑥𝑆𝐴 )2 + (𝑐 − 𝑦𝑆𝐴 )2 + 𝑞𝐴 2 − 𝐿2 ]=0 +[ 𝑏2 + 𝑑 2 + 1.
(97) 97. 𝑝= 𝑞=. 2𝑏(𝑎 − 𝑥𝑆𝐴 ) + 2𝑑 (𝑐 − 𝑦𝑆𝐴) − 2𝑞𝐴 𝑏2 + 𝑑 2 + 1. (𝑎 − 𝑥𝑆𝐴 )2 + (𝑐 − 𝑦𝑆𝐴)2 + 𝑞𝐴 2 − 𝐿2 𝑏2 + 𝑑 2 + 1. 𝑤 2 + 𝑝𝑤 + 𝑞 = 0. 𝑤1,2. 𝑝 𝑝2 √ =− ± −𝑞 2 4. 𝑝2 Δ= −𝑞 4. 𝑤1 𝛥>0. 𝛥=0 𝑆𝐴 , 𝑆𝐵 𝛥<0. 𝑆𝐶. 𝑤2.
(98) 98.
(99) 99. 𝑞𝐴 𝑞𝐵. 𝑞𝐶. ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗𝐶𝑖 + ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑂𝑃 + ⃗⃗⃗⃗⃗⃗⃗ 𝑃𝐻𝑖 = 𝑂𝐴 𝐴𝐶𝑖 𝑄𝑖 + ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄𝑖 𝐻𝑖 ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄 𝑖 𝐻𝑖. 𝑄𝑖. 𝐻𝑖 𝑄𝑖. 𝑄𝑖. 𝐻𝑖. 𝐻𝑖.
(100) 100. ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄 𝑖 𝐻𝑖 ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ 𝐿𝑖 = 𝑄 𝑖 𝐻𝑖 𝑄𝑖 = (𝑥𝑄𝑖 , 𝑦𝑄𝑖, 𝑧𝑄𝑖, ) = (𝑥𝐴𝑐𝑖 , 𝑦𝐴𝑐𝑖, , 𝑞𝑖 ). 𝐻𝑖 = (𝑥𝐻𝑖 , 𝑦𝐻𝑖, 𝑧𝐻𝑖, ) = (𝑢 + ℎ𝑥𝑖 , 𝑣 + ℎ𝑦𝑖 , 𝑤 ). ⃗⃗⃗𝑖 = ⃗⃗⃗⃗⃗⃗⃗⃗ 𝐿 𝑄𝑖 𝐻𝑖 = 𝐻𝑖 − 𝑄𝑖 = [𝑢 + ℎ𝑥𝑖 − 𝑥𝐴𝑐𝑖 , 𝑣 + ℎ𝑦𝑖 − 𝑦𝐴𝑐𝑖 , 𝑤 − 𝑞𝑖 ]. 𝑣𝑝 = [𝑢̇ , 𝑣̇ , 𝑤̇ ]𝑇 ⃗⃗⃗⃗. 𝑣𝑄𝑖 = [0, 0, 𝑞𝑖̇ ] ⃗⃗⃗⃗⃗. 𝑇. ⃗⃗⃗⃗⃗⃗⃗ 𝑃𝐻𝑖. ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑂𝐴𝐶𝑖. ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑑𝑂𝑃 𝑑𝐴 𝑑𝑄 𝐶𝑖 𝑄𝑖 𝑖 𝐻𝑖 = ⃗⃗⃗⃗ 𝑣𝑝 = + 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑝 𝑣𝑝 = ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ 𝑣𝑄𝑖 + 𝑣 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⁄. 𝑄𝑖. ⃗⃗⃗⃗𝑝 𝑣 𝑣𝑄𝑖 ⃗⃗⃗⃗⃗ 𝑝 𝑣 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⁄. 𝑄𝑖. 𝑃. ⃗⃗⃗𝑖 𝑝 𝑣 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗⃗ 𝑤34 ×𝐿 ⁄ 𝑄𝑖. 𝑄𝑖.
(101) 101. ⃗⃗⃗𝑖 𝑣𝑝 = ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ 𝑣𝑄𝑖 + ⃗⃗⃗⃗⃗⃗⃗ 𝑤34 ×𝐿 ⃗⃗⃗ 𝐿𝑖. ⃗⃗⃗ ⃗⃗⃗𝑝 = 𝐿 ⃗⃗⃗𝑖 ∙ ⃗⃗⃗⃗⃗ ⃗⃗⃗ 𝐿𝑖 ∙ 𝑉 𝑣𝑄𝑖 + ⃗⃗⃗ 𝐿𝑖 ∙ (𝑤 ⃗⃗⃗⃗⃗⃗⃗ 34 ×𝐿𝑖 ) ⃗⃗⃗ ⃗⃗⃗ 𝐿𝑖 ∙ (𝑤 ⃗⃗⃗⃗⃗⃗⃗ 34 ×𝐿𝑖 ) = 0. ⃗⃗⃗ ⃗⃗⃗𝑖 ∙ ⃗⃗⃗⃗⃗ 𝐿𝑖 ∙ ⃗⃗⃗⃗ 𝑣𝑝 = 𝐿 𝑣𝑄𝑖. ⃗⃗⃗ 𝐿𝑖 ∙ ⃗⃗⃗⃗ 𝑣𝑝 = [𝑢 + ℎ𝑥𝑖 − 𝑥𝐴𝑐𝑖 , 𝑣 + ℎ𝑦𝑖 − 𝑦𝐴𝑐𝑖 , 𝑤 − 𝑞𝑖 ] ∙ [𝑢̇ , 𝑣̇ , 𝑤̇ ] ⃗⃗⃗ 𝐿𝑖 ∙ 𝑣 ⃗⃗⃗⃗⃗ 𝑄𝑖 = [𝑢 + ℎ𝑥𝑖 − 𝑥𝐴𝑐𝑖 , 𝑣 + ℎ𝑦𝑖 − 𝑦𝐴𝑐𝑖 , 𝑤 − 𝑞𝑖 ] ∙ [0, 0, 𝑞𝑖̇ ]. 𝑢 + ℎ𝑥𝐴 − 𝑥𝐴𝑐𝐴 [𝑢 + ℎ𝑥𝐵 − 𝑥𝐴𝑐𝐵 𝑢 + ℎ𝑥𝐶 − 𝑥𝐴𝑐𝐶. 𝑣 + ℎ𝑦𝐴 − 𝑦𝐴𝑐𝐴 𝑤 − 𝑞𝐴 𝑢̇ 𝑣 + ℎ𝑦𝐵 − 𝑦𝐴𝑐𝐵 𝑤 − 𝑞𝐵 ] { 𝑣̇ } 𝑣 + ℎ𝑦𝐶 − 𝑦𝐴𝑐𝐶 𝑤 − 𝑞𝐶 𝑤̇ 𝑞𝐴̇ 𝑤 − 𝑞𝐴 0 0 𝑤 − 𝑞𝐵 0 ] {𝑞𝐵̇ } =[ 0 0 0 𝑤 − 𝑞𝐶 𝑞𝐶̇. [𝐽𝑥 ]{⃗⃗⃗⃗ 𝑣𝑃 } = [𝐽𝑞 ]{𝑣 ⃗⃗⃗⃗⃗ 𝑄𝑖 } [𝐽𝑥 ] [𝐽𝑞 ] [𝐽𝑞 ]. −1.
(102) 102. −1. [𝐽𝑞 ] [𝐽𝑥 ]{⃗⃗⃗⃗ 𝑣𝑃 } = {𝑣 ⃗⃗⃗⃗⃗ 𝑄𝑖 }. −1. [𝐽] = [𝐽𝑞 ]. ∙ [𝐽𝑥 ]. [𝐽 ] ∆𝑧1 (∆𝑦2 ∆𝑧3 − ∆𝑦3 ∆𝑧2) ∆𝑧2 (∆𝑥2 ∆𝑧3 − ∆𝑥3 ∆𝑧2 ) ∆𝑧3 (∆𝑥2 ∆𝑦3 − ∆𝑥3 ∆𝑦2 ) − ∆0 ∆0 ∆0 ∆𝑧1 (∆𝑦1 ∆𝑧3 − ∆𝑦3 ∆𝑧1) ∆𝑧2 (∆𝑥1 ∆𝑧3 − ∆𝑥3 ∆𝑧1 ) ∆𝑧3(∆𝑥1 ∆𝑦3 − ∆𝑥3 ∆𝑦1 ) 𝐽= − − ∆0 ∆0 ∆0 ∆𝑧1 (∆𝑦1∆𝑧2 − ∆𝑦2 ∆𝑧1) ∆𝑧2 (∆𝑥1 ∆𝑧2 − ∆𝑥2 ∆𝑧1 ) ∆𝑧3(∆𝑥1 ∆𝑦2 − ∆𝑥2 ∆𝑦1 ) − [ ] ∆0 ∆0 ∆0. ∆𝑥𝑖 = (𝑃𝑥 + ℎ𝑥𝑖 − 𝑥𝑖 ). ∆𝑧𝑖 = (𝑃𝑧 − 𝑎𝑖 ). ∆𝑦𝑖 = (𝑃𝑦 + ℎ𝑦𝑖 − 𝑦𝑖 ). ∆0 = ∆𝑥1 ∆𝑦2 ∆𝑧3 − ∆𝑥1 ∆𝑦3 ∆𝑧2 − ∆𝑥2 ∆𝑦1∆𝑧3 + ∆𝑥2 ∆𝑦3 ∆𝑧1 + ∆𝑥3 ∆𝑦1 ∆𝑧2 − ∆𝑥3 ∆𝑦2 ∆𝑧1 [𝐽𝑥 ]−1 {⃗⃗⃗⃗ 𝑣𝑃 } = [𝐽𝑥 ]−1 [𝐽𝑞 ]{𝑣 ⃗⃗⃗⃗⃗ 𝑄𝑖 }. [𝐽 ]−1 = [𝐽𝑥 ]−1 [𝐽𝑞 ]. 𝑥̇ ∙ (𝑥 − 𝑥𝑆𝑖 ) + 𝑦̇ ∙ (𝑦 − 𝑦𝑆𝑖 ) + 𝑧̇ ∙ (𝑧 − 𝑧𝑆𝑖 ) − 𝑧𝑆𝑖̇ ∙ (𝑧 − 𝑧𝑆𝑖 ) = 0 𝑧𝑆𝐴 , 𝑧𝑆𝐵 , 𝑧𝑆𝐶. 𝑞𝐴 𝑞𝐵 𝑞𝐶. 𝑢̇ ∙ (𝑢 − 𝑥𝑆𝑖 ) + 𝑣̇ ∙ (𝑣 − 𝑦𝑆𝑖 ) + 𝑤̇ ∙ (𝑤 − 𝑞𝑖 ) − 𝑞𝑖̇ ∙ (𝑤 − 𝑞𝑖 ) = 0.
(103) 103. 𝑢 − 𝑥𝑆𝐴 [𝑢 − 𝑥𝑆𝐵 𝑢 − 𝑥𝑆𝐶. 𝑣 − 𝑦𝑆𝐴 𝑣 − 𝑦𝑆𝐵 𝑣 − 𝑦𝑆𝐶. 𝑤 − 𝑞𝐴 𝑢̇ 𝑤 − 𝑞𝐴 𝑤 − 𝑞𝐵 ] { 𝑣̇ } = [ 0 𝑤 − 𝑞𝐶 𝑤̇ 0. 𝑤 − 𝑞𝐴 [𝐽𝑞 ] = [ 0 0 𝑢 − 𝑥𝑆𝐴 [𝐽𝑥 ] = [𝑢 − 𝑥𝑆𝐵 𝑢 − 𝑥𝑆𝐶. 0 𝑤 − 𝑞𝐵 0. 𝑣 − 𝑦𝑆𝐴 𝑣 − 𝑦𝑆𝐵 𝑣 − 𝑦𝑆𝐶. 0 𝑤 − 𝑞𝐵 0. 0 0 ] 𝑤 − 𝑞𝐶. ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑃 − 𝑆𝑆𝐴 ) 𝑤 − 𝑞𝐴 𝑤 − 𝑞𝐵 ] = [(𝑃 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ − 𝑆𝑆𝐵 )] 𝑤 − 𝑞𝐶 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑃 − 𝑆𝑆𝐶 ). (𝑢 − 𝑥𝑆𝐴 ) (𝑣 − 𝑦𝑆𝐴 ) 𝑤 − 𝑞𝐴 𝑤 − 𝑞𝐴 (𝑢 − 𝑥𝑆𝐵 ) (𝑣 − 𝑦𝑆𝐵 ) 𝐽= 𝑤 − 𝑞𝐵 𝑤 − 𝑞𝐵 (𝑢 − 𝑥𝑆𝐶 ) (𝑣 − 𝑦𝑆𝐶 ) [ 𝑤 − 𝑞𝐶 𝑤 − 𝑞𝐶. (𝑥𝑆𝐴 , 𝑦𝑆𝐴) = (0 , 𝐿𝑅𝑟 ). (𝑥𝑆𝐵 , 𝑦𝑆𝐵 ) = (. 𝑞𝐴̇ 0 0 ] {𝑞𝐵̇ } 𝑤 − 𝑞𝐶 𝑞𝐶̇. 1 1 1 ]. 1 √3 𝐿𝑅𝑟 , − 𝐿𝑅𝑟 ) 2 2. 𝑎 ⃗⃗⃗⃗𝑝 = [𝑢̈ , 𝑣̈ , 𝑤̈ ]𝑇. (𝑥𝑆𝐶 , 𝑦𝑆𝐶 ) = (−. 1 √3 𝐿𝑅𝑟 , − 𝐿𝑅𝑟 ) 2 2.
(104) 104. 𝑎𝑄𝑖 = [0, 0, 𝑞𝑖̈ ]𝑇 ⃗⃗⃗⃗⃗. ⃗⃗⃗⃗ 𝑑 𝑑𝑃⃗ 𝑑 𝑑 𝑑𝑄𝑖 𝑑 [𝐽𝑥 ] ∙ ∙ {𝑣𝑝 } + [𝐽𝑥 ] ∙ {𝑣𝑝 } = [𝐽𝑞 ] ∙ ∙ {𝑣𝑄𝑖 } + [𝐽𝑞 ] ∙ {𝑣𝑄𝑖 } 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡. {𝑣𝑝 } = {𝑎𝑝 } =. 𝑑 {𝑣𝑝 } = [𝑢̈ , 𝑣̈ , 𝑤̈ ]𝑇 𝑑𝑡. {𝑣𝑄𝑖 } = {𝑣𝑄𝑖 } =. [𝐽𝑥̇ ] ∙ {𝑣𝑝 } ∙ {𝑣𝑝 } + [𝐽𝑥 ]. 𝑑𝑃⃗ = [𝑢̇ , 𝑣̇ , 𝑤̇ ]𝑇 𝑑𝑡. ⃗⃗⃗⃗ 𝑑𝑄𝑖 𝑇 = [0, 0, 𝑞𝑖̇ ] 𝑑𝑡. 𝑑 {𝑣 } = [0, 0, 𝑞𝑖̈ ]𝑇 𝑑𝑡 𝑄𝑖 [𝐽𝑥̇ ] =. 𝑑 [𝐽 ] 𝑑𝑡 𝑥. [𝐽𝑞̇ ] =. 𝑑 [𝐽 ] 𝑑𝑡 𝑞. 𝑑 𝑑 {𝑣𝑝 } = [𝐽𝑞̇ ] ∙ {𝑣𝑎 } ∙ {𝑣𝑎 } + [𝐽𝑞 ] {𝑣𝑎 } 𝑑𝑡 𝑑𝑡. 𝑢̇ 2 𝑞𝐴̇ 2 𝑞𝐴̈ 𝑢̈ [𝐽𝑥̇ ] ∙ { 𝑣̇ 2 } + [𝐽𝑥 ] ∙ { 𝑣̈ } = [𝐽𝑞̇ ] ∙ {𝑞𝐵̇ 2 } + [𝐽𝑞 ] ∙ {𝑞𝐵̈ } 𝑤̈ 𝑞𝐶̈ 𝑤̇ 2 𝑞𝐶̇ 2 [𝐽𝑞 ]. −1.
(105) 105. 𝑢̇ 2 𝑞𝐴̇ 2 𝑞𝐴̈ 𝑢̈ −1 −1 −1 2 2 ̇ ̇ [𝐽𝑞 ] ∙ [𝐽𝑥 ] ∙ { 𝑣̈ } + [𝐽𝑞 ] ∙ [𝐽𝑥 ] ∙ { 𝑣̇ } − [𝐽𝑞 ] ∙ [𝐽𝑞 ] ∙ {𝑞𝐵̇ } = {𝑞𝐵̈ } 𝑤̈ 𝑞𝐶̈ 𝑤̇ 2 𝑞̇ 2 𝐶. −1. [𝐽] = [𝐽𝑞 ]. ∙ [𝐽𝑥 ]. −1. ∙ [𝐽𝑥̇ ]. −1. ∙ [𝐽𝑞̇ ]. [𝐾𝑞𝑥 ] = [𝐽𝑞 ] [𝐾𝑞𝑞 ] = [𝐽𝑞 ]. 𝑢̇ 2 𝑞𝐴̇ 2 𝑞𝐴̈ 𝑢̈ [𝐽] ∙ { 𝑣̈ } + [𝐾𝑞𝑥 ] { 𝑣̇ 2 } − [𝐾𝑞𝑞 ] {𝑞𝐵̇ 2 } = {𝑞𝐵̈ } 𝑤̈ 𝑞𝐶̈ 𝑤̇ 2 𝑞̇ 2 𝐶. [𝐽𝑥 ]−1 𝑞𝐴̇ 2 𝑢̇ 2 𝑞𝐴̈ 𝑢̈ { 𝑣̈ } = [𝐽𝑥 −1 ] ∙ [𝐽𝑞 ] ∙ {𝑞𝐵̈ } + [𝐽𝑥 −1 ] ∙ [𝐽𝑞̇ ] ∙ {𝑞𝐵̇ 2 } − [𝐽𝑥 −1 ] ∙ [𝐽𝑥̇ ] ∙ { 𝑣̇ 2 } 𝑤̈ 𝑞𝐶̈ 𝑞̇ 2 𝑤̇ 2 𝐶. [𝐽 ]−1 = [𝐽𝑥 ]−1 ∙ [𝐽𝑞 ] [𝐾𝑥𝑞 ] = [𝐽𝑥 ]−1 ∙ [𝐽𝑞̇ ] [𝐾𝑥𝑥 ] = [𝐽𝑥 ]−1 ∙ [𝐽𝑥̇ ]. 𝑞𝐴̇ 2 𝑢̇ 2 𝑞𝐴̈ 𝑢̈ { 𝑣̈ } = [𝐽 ]−1 ∙ {𝑞𝐵̈ } + [𝐾𝑥𝑞 ] ∙ {𝑞𝐵̇ 2 } − [𝐾𝑥𝑥 ] ∙ { 𝑣̇ 2 } 𝑤̈ 𝑞𝐶̈ 𝑞̇ 2 𝑤̇ 2 𝐶.
(106) 106. 𝑢̇ , 𝑣̇ , 𝑤̇ 𝑞̇ A , 𝑞̇ B , 𝑞̇ C. 𝑞̇ A , 𝑞̇ B 𝑢̇ , 𝑣̇. 𝑤̇. 𝑢̇ , 𝑣̇ 𝑞̇ A , 𝑞̇ B. 𝑞̇ C. 𝑤̇. 𝑞̇ C.
(107) 107. det(𝐽𝑞 ) = 0. {𝑤|𝑤 = 𝑞𝐴 } ∪ {𝑤|𝑤 = 𝑞𝐵 } ∪ {𝑤|𝑤 = 𝑞𝐶 }. 𝐿 = 𝐿𝑅𝑟 = 𝑅 − 𝑟. 𝐿 = 460 mm 𝐿𝑅𝑟 < 𝐿. 𝐿𝑅𝑟 = 250 mm.
(108) 108. det(𝐽𝑥 ) = 0.
(109) 109. 𝑢 − 𝑥𝑆𝐴 𝑣 − 𝑦𝑆𝐴 𝑤 − 𝑞𝐴 𝑢 − 𝑥 𝑣 − 𝑦 𝑤 𝑐1 ∙ { 𝑆𝐵 } + 𝑐2 ∙ { 𝑆𝐵 } + 𝑐3 ∙ { − 𝑞𝐵 } = 0 𝑢 − 𝑥𝑆𝐶 𝑣 − 𝑦𝑆𝐶 𝑤 − 𝑞𝐶 𝑐1 ∙ {𝑗X1} + 𝑐2 ∙ {𝑗X2 } + 𝑐3 ∙ {𝑗X3} = 0. 𝑗x1. 𝑗x2 𝑗x3. 𝑤 − 𝑞𝐴 = 𝑤 − 𝑞𝐵 = 𝑤 − 𝑞𝐶 = 0 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑃 − 𝑆𝑆𝑖 ). ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑃 − 𝑆𝑆𝑖 ) = ±⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑃 − 𝑆𝑆𝑘 ). 𝑖≠𝑘.
(110) 110. 𝝉 = [𝜏1, 𝜏2 , 𝜏3 , ⋯ , 𝜏𝑛 ]𝑇 ∆𝒒 = [∆𝑞1 , ∆𝑞2 , ∆𝑞3 , ⋯ , ∆𝑞𝑛 ]𝑇 𝝉 ∆𝒒 𝝉 = 𝜒∆𝒒 𝜒 = diag[𝑘1 , 𝑘2 , 𝑘3 , ⋯ , 𝑘𝑛 , ].
(111) 111. 𝑭 = 𝐽t 𝝉 𝐽t ∆𝑥 = [∆𝑥, ∆𝑦, ∆𝑧, ∆𝜙, ∆𝜃, ∆𝜓]𝑇. ∆𝒒 = 𝐽∆𝒙. 𝑭 = 𝐾∆𝒙 𝐾 = 𝐽t 𝜒 𝐽. 𝑘a = 𝑘b = 𝑘c = 𝑘act. 𝐾 = 𝐽T (𝑘act ∙ 𝐼 ) 𝐽 = 𝑘act ∙ 𝐽T 𝐽. 𝑘act = 1 kN/m 𝑘act = 106 kN/m 𝑘act = 105 kN/m 𝑘act = 105 kN/m.
(112) 112. Δ𝑈Sa Δ𝑊Sa Δ𝑈Sb 𝐽= Δ𝑊Sb Δ𝑈Sc [ Δ𝑊Sc. Δ𝑈S𝑖 = 𝑢 − 𝑥S𝑖. Δ𝑉Sa Δ𝑊Sa Δ𝑉Sb Δ𝑊Sb Δ𝑉Sc Δ𝑊Sc. 1 1 1 ]. Δ𝑉S𝑖 = 𝑣 − 𝑦S𝑖. Δ𝑈Sa Δ𝑊Sa T 𝐽 = Δ𝑉Sa Δ𝑊Sa [ 1. Δ𝑈Sb Δ𝑊Sb Δ𝑉Sb Δ𝑊Sb 1. 𝐾11 𝑲 = 𝑘Act [ 𝐾21 𝐾31. 𝐾11 = 𝑘Act ∙ [(. Δ𝑈Sa Δ𝑊Sa. 𝐾22 = 𝑘Act ∙ [(. Δ𝑉Sa Δ𝑊Sa. 𝐾12 𝐾22 𝐾32. 2. ) +(. Δ𝑈Sb Δ𝑊Sb. 2. ) +(. Δ𝑉Sb Δ𝑊Sb. 𝐾33 = 3 ∙ 𝑘Act. Δ𝑊S𝑖 = 𝑤 − 𝑞𝑖. Δ𝑈Sc Δ𝑊Sc Δ𝑉Sc Δ𝑊Sc 1 ]. 𝐾13 𝐾23 ] 𝐾33. 2. ) +( 2. ) +(. Δ𝑈Sc Δ𝑊Sc Δ𝑉Sc Δ𝑊Sc. 2. )] 2. )].
(113) 113. 𝐾12 = 𝐾21 = 𝑘Act ∙ [. Δ𝑈Sa Δ𝑉Sa. (Δ𝑊Sa ). 𝐾13 = 𝐾31 = 𝑘Act ∙ [. Δ𝑊Sa Δ𝑉Sa Δ𝑊Sa. Δ𝑈Sa Δ𝑊Sa. 𝐾Sty = 𝐾22 = 𝑘Act ∙ [(. +. Δ𝑈Sa. 𝐾23 = 𝐾32 = 𝑘Act ∙ [. 𝐾Stx = 𝐾11 = 𝑘Act ∙ [(. 2. Δ𝑉Sa Δ𝑊Sa. Δ𝑈Sb Δ𝑉Sb. (Δ𝑊Sb ) Δ𝑈Sb. +. Δ𝑉Sb Δ𝑊Sb. 2. ) +(. +. Δ𝑊Sb. +. Δ𝑈Sb Δ𝑊Sb. 2. ) +(. Δ𝑉Sb Δ𝑊Sb. 2. +. +. Δ𝑈Sc Δ𝑉Sc. (Δ𝑊Sc )2. Δ𝑈Sc Δ𝑊Sc Δ𝑉Sc Δ𝑊Sc. 2. ) +(. ] ]. Δ𝑈Sc Δ𝑊Sc. 2. ) +(. Δ𝑉Sc Δ𝑊Sc. 𝐾Stz = 𝐾33 = 3 ∙ 𝑘Act. 𝐾Stx 𝐾Sty 𝐾Stz. 𝐾𝑆𝑇𝑇 𝐾𝑆𝑇𝑇 = 𝑡𝑟(𝐾 ) = 𝐾𝑆𝑇𝑋 + 𝐾𝑆𝑇𝑌 + 𝐾𝑆𝑇𝑍. ]. 2. )] 2. )].
(114) 114. 𝐾𝑆𝑇𝑇 = 𝑘𝐴𝐶𝑇 ∙ {[(. Δ𝑈𝑆𝐴. 2. ) +(. Δ𝑊𝑆𝐴 Δ𝑉𝑆𝐴. + [(. Δ𝑊𝑆𝐴. 2. ). Δ𝑈𝑆𝐵. 2. Δ𝑈𝑆𝐶. 2. ) +( )] Δ𝑊𝑆𝐵 Δ𝑊𝑆𝐶 Δ𝑉𝑆𝐵 2 Δ𝑉𝑆𝐶 2 ) +( ) ] + 3} +( Δ𝑊𝑆𝐵 Δ𝑊𝑆𝐶.
(115) 115.
(116) 116.
(117) 117. 𝐶𝐼 =. 1 𝜅. 𝜅. 𝜅 = ‖𝐽−1 ‖‖𝐽‖. 𝜅=√. 𝜆1 𝜆2. 𝜆1 𝜆2.
(118) 118.
(119) 119.
(120) 120. 𝑃𝑥 = 0. 𝑃𝑦 = 0. 𝑃𝑧.
(121) 121.
(122) 122. 𝑞A 𝑞B 𝑞C.
(123) 123. 𝑂 𝑂. 𝑏𝐺3𝑖 = 𝑂𝑏𝐶𝑖 + 𝐶𝑖 𝑏𝐺3𝑖. 𝑏𝐺4𝑖 = 𝑂𝑏𝐷𝑖 + 𝐷𝑖𝑏𝐺4𝑖.
(124) 124. 𝑂. 𝑏𝐺3𝑖. 𝑂. 𝑏𝐺4𝑖. 𝑂. 𝑂. 𝑇. 𝑂. 𝑏𝐺3𝑖 = [𝑏𝐺3𝑖𝑥 , 𝑏𝐺3𝑖𝑦 , 𝑏𝐺3𝑖𝑧 ]. 𝑂. 𝑏𝐺4𝑖 = [𝑏𝐺4𝑖𝑥 , 𝑏𝐺4𝑖𝑦 , 𝑏𝐺4𝑖𝑧 ]. 𝑇. 𝑏𝐶𝑖. 𝑏𝐷𝑖 𝑇. 𝑏𝐶𝑖. 𝐶𝑖 𝐷𝑖 = [𝑎𝑖𝑥 , 𝑎𝑖𝑦 + , 𝑞𝑖 ] 2. 𝑏𝐷𝑖. 𝐶𝑖 𝐷𝑖 = [𝑎𝑖𝑥 , 𝑎𝑖𝑦 − , 𝑞𝑖 ] 2. 𝑂. 𝑂. 𝑇. 𝐶𝑖 𝐷𝑖 𝐶𝑖. 𝑏𝐺3𝑖. 𝐷𝑖. 𝑏𝐺3𝑖 𝐶𝑖. 𝑏𝐺3𝑖 = 𝐷𝑖 𝑏𝐺4𝑖 = 𝑄𝑖. 𝑄𝑖. 𝑇𝐶´𝑖 𝐶´𝑖 𝑏𝐺3𝑖 𝑄𝑖 𝑇𝐶´𝑖 𝐷´𝑖 𝑏𝐺4𝑖. 𝑇𝐶´𝑖 cos(𝜌𝑖 )cos(𝛾𝑖 ) 𝑄𝑖. 𝐶´𝑖. 𝑏𝐺𝑖. 𝐶´𝑖. 𝑏𝐺𝑖. 𝑇𝐶´𝑖 = sin(𝜌𝑖 )cos(𝛾𝑖 ) −𝑠𝑖𝑛(𝛾𝑖 ) [ 0. 𝐶´𝑖. −sin(𝜌𝑖 ). cos(𝜌𝑖 )sin(𝛾𝑖 ). cos(𝜌𝑖 ). sin(𝜌𝑖 )cos(𝛾𝑖 ). 0 0. cos(𝛾𝑖 ) 0. 𝑏𝐺3𝑖 = 𝑒𝑖 ∙ 𝐶´𝑖 𝑠𝑖 𝐷´𝑖 𝑏𝐺4𝑖 = 𝑒𝑖 ∙ 𝐷´𝑖 𝑠𝑖 𝑒𝑖. 0 ±. 𝐶𝑖 𝐷𝑖 2 0 1 ].
(125) 125. 𝑒𝑖 =. 𝑙𝑖 2. 𝑙𝑖 𝐶´𝑖 𝑠 𝑖 𝐷´𝑖 𝑠 𝑖 𝐶´𝑖. 𝑠𝑖 = [1, 0, 0]𝑇. 𝐷´𝑖. 𝑠𝑖 = [1, 0, 0]𝑇 ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄 𝑖 𝐻𝑖. 𝐶𝑖. 𝑏𝐺3𝑖 =. 𝐷𝑖. 𝑏𝐺4𝑖 =. 𝑄𝑖. 𝑏𝐺𝑖. 𝑣𝐻𝑖 = 𝑣𝑄𝑖 + 𝜔𝑖 × 𝑄𝑖𝑏𝐻𝑖. 𝒗𝐸𝐼 𝒗𝐻𝑖 = 𝒗𝑃 = [𝑢̇ , 𝑣̇ , 𝑤̇]𝑇 𝒗𝐶𝐼 𝒗𝑄𝑖 = [0,0, 𝑞̇ 𝑖 ]𝑇 𝑄𝑖. 𝑏𝐻𝑖.
(126) 126. 𝐻𝑖 = (𝑥𝐻𝑖 , 𝑦𝐻𝑖, 𝑧𝐻𝑖, ) = (𝑢 + ℎ𝑥𝑖 , 𝑣 + ℎ𝑦𝑖 , 𝑤 ) ; 𝑄𝑖 = (𝑥𝑄𝑖 , 𝑦𝑄𝑖, 𝑧𝑄𝑖, ) = (𝑥𝐴𝑐𝑖 , 𝑦𝐴𝑐𝑖, , 𝑞𝑖 ) 𝑄𝑖. 𝑏𝐻𝑖 = ⃗⃗⃗ 𝐿𝑖 = ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄𝑖 𝐻𝑖 = [𝑢 + ℎ𝑥𝑖 − 𝑥𝐴𝑐𝑖 , 𝑣 + ℎ𝑦𝑖 − 𝑦𝐴𝑐𝑖 , 𝑤 − 𝑞𝑖 ]. 𝑄𝑖. ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑏𝐻𝑖 = ⃗⃗⃗ 𝐿𝑖 = 𝑄 𝑖 𝐻𝑖 = [Δ𝑈S𝑖 , Δ𝑉S𝑖 , Δ𝑊S𝑖 ]. Δ𝑈S𝑖 = 𝑢 − (𝑥𝐴𝑐𝑖 − ℎ𝑥𝑖 ). Δ𝑉S𝑖 = 𝑣 − (𝑦𝐴𝑐𝑖 − ℎ𝑦𝑖 ). ̂ 𝑖 𝑄𝑖𝑏𝐻𝑖 = 𝜔𝑖 × 𝑄𝑖𝑏𝐻𝑖 𝝎. ω ̂𝑖 0 ̂ 𝑖 = [ 𝜔𝑧 𝝎 −𝜔𝑦. 0 0 𝑢̇ [ 𝑣̇ ] = [ 0 ] + [ 𝜔𝑧 𝑞̇ 𝑖 −𝜔𝑦 𝑤̇ 0 𝑢̇ [ 𝑣̇ ] = [ 𝜔𝑧 𝑤̇ − 𝑞̇ 𝑖 −𝜔𝑦. −𝜔𝑧 0 𝜔𝑥. 𝜔𝑦 −𝜔𝑥 ] 0. −𝜔𝑧 0 𝜔𝑥 −𝜔𝑧 0 𝜔𝑥. 𝜔𝑦 Δ𝑈S𝑖 −𝜔𝑥 ] [ Δ𝑉S𝑖 ] 0 Δ𝑊S𝑖 𝜔𝑦 Δ𝑈S𝑖 −𝜔𝑥 ] [ Δ𝑉S𝑖 ] 0 Δ𝑊S𝑖. ̅̅̅̅̅̅ 𝑄𝑖̇ 𝐻𝑖̇ 𝜔𝑖 ∙ 𝑄𝑖 𝑏𝐻𝑖 = 0 𝜔𝑥 Δ𝑈S𝑖 + 𝜔𝑦 Δ𝑉S𝑖 + 𝜔𝑧 Δ𝑊S𝑖 = 0. Δ𝑊S𝑖 = 𝑤 − 𝑞𝑖.
(127) 127. 0 [Δ𝑈S𝑖 Δ𝑉S𝑖. 𝜔𝑥 = (. Δ𝑊S𝑖 Δ𝑉S𝑖 −Δ𝑈S𝑖. −Δ𝑉S𝑖 𝜔𝑥 𝑢̇ 𝜔 Δ𝑊S𝑖 ] [ 𝑦 ] = [ 0 ] 𝜔𝑧 𝑤̇ − 𝑞̇ 𝑖 0. Δ𝑈S𝑖 2 1 Δ𝑈S𝑖 2 1 ) ( ) ∙ 𝑢̇ + [1 − ( ) ]∙( ) ∙ (𝑤̇ − 𝑞𝑖̇ ) 𝑙 Δ𝑉S𝑖 𝑙 Δ𝑉S𝑖 𝜔𝑦 =. Δ𝑊S𝑖 Δ𝑈S𝑖 (𝑤̇ − 𝑞𝑖̇ ) 𝑢̇ − 𝑙2 𝑙2. Δ𝑈S𝑖 Δ𝑊S𝑖 Δ𝑊S𝑖 2 1 ( ) ) ∙ ( 2 ) 𝑤̇ − 𝑞𝑖̇ + [( ) − 1] 𝜔𝑧 = − ( 𝑢̇ Δ𝑉S𝑖 𝑙 𝑙 Δ𝑉S𝑖. 𝑢̇ =. Δ𝑈S𝐴 Δ𝑉S𝐴 𝑞̇ 𝐴 + 𝑞̇ + 𝑞̇ 𝐶 Δ𝑊S𝐴 Δ𝑊S𝐴 𝐵. 𝑤̇ =. Δ𝑈S𝐶 Δ𝑉S𝐶 𝑞̇ 𝐴 + 𝑞̇ + 𝑞̇ 𝐶 Δ𝑊S𝐶 Δ𝑊S𝐶 𝐵. 𝜔𝑥𝑖 𝑐11 1 {[𝑐21 𝜔𝑖 = [𝜔𝑦𝑖 ] = (Δ𝑉Si )𝑙 2 𝑐 𝜔𝑧𝑖 31. 𝑐12 𝑐22 𝑐32. 𝑐13 𝑞̇ 𝐴 0 0 𝑑13 0 𝑐23 ] [𝑞̇ 𝐵 ] + [0 0 𝑑23 ] [ 0 ]} 𝑐33 𝑞̇ 𝐶 0 0 𝑑33 𝑞̇ 𝑖. 𝑐11 =. Δ𝑈Si Δ𝑊Si Δ𝑈S𝐴 Δ𝑈Sc + [𝑙 2 − (Δ𝑈Si )2 ] Δ𝑊S𝐴 Δ𝑊S𝐶. 𝑐12 =. Δ𝑈Si Δ𝑊Si Δ𝑉S𝐴 Δ𝑉Sc + [𝑙 2 − (Δ𝑈Si )2 ] Δ𝑊S𝐴 Δ𝑊S𝐶. 𝑐13 = Δ𝑈Si Δ𝑊Si + [𝑙 2 − (Δ𝑈Si )2 ].
(128) 128. 𝑐21 =. Δ𝑉SiΔ𝑊Si Δ𝑈S𝐴 Δ𝑈Si Δ𝑉Si Δ𝑈S𝐶 − Δ𝑊S𝐴 Δ𝑊S𝐶. 𝑐22 =. Δ𝑉Si Δ𝑊SiΔ𝑉S𝐴 Δ𝑈Si Δ𝑉Si Δ𝑉S𝐶 − Δ𝑊S𝐴 Δ𝑊S𝐶. 𝑐23 = Δ𝑉Si Δ𝑊Si − Δ𝑈Si Δ𝑉Si 𝑐31 = [(Δ𝑊Si )2 − 𝑙 2 ]. Δ𝑈S𝐴 Δ𝑈Si Δ𝑊Si Δ𝑈S𝐶 − Δ𝑊S𝐴 Δ𝑊S𝐶. 𝑐32 = [(Δ𝑊Si )2 − 𝑙 2 ]. Δ𝑉S𝐴 Δ𝑈Si Δ𝑊Si Δ𝑉S𝐶 − Δ𝑊S𝐴 Δ𝑊S𝐶. 𝑐33 = [(Δ𝑊Si )2 − 𝑙 2 ] − Δ𝑈Si Δ𝑊Si 𝑑13 = [(Δ𝑈Si )2 − 𝑙 2 ] 𝑑23 = Δ𝑈Si Δ𝑉Si 𝑑33 = Δ𝑈Si Δ𝑊Si. 𝑣𝐺3𝑖 = 𝑣𝐶𝑖 + 𝜔𝑖 × 𝐶𝑖𝑏𝐺3𝑖 𝑣𝐺4𝑖 = 𝑣𝐷𝑖 + 𝜔𝑖 × 𝐷𝑖𝑏𝐺4𝑖. 𝑣𝐺3𝑖 𝑣𝐺4𝑖 𝑣𝐶𝑖 𝑣𝐷𝑖 Ω𝑖 𝐶𝑖. 𝑏𝐺3𝑖. 𝐷𝑖. 𝑏𝐺4𝑖.
(129) 129. 𝑣𝐷𝑖 = 𝑣𝐶𝑖 = 𝑣𝑄𝑖. 𝑣𝐺3𝑖 = 𝑣𝐺4𝑖 = 𝑣𝐺𝑖 𝑣𝐺𝑖 = 𝑣𝑄𝑖 + 𝜔𝑖 × 𝑄𝑖𝑏𝐺𝑖. 𝑄𝑖. 𝑏𝐺𝑖 =. ⃗⃗⃗ 𝐿𝑖 Δ𝑈S𝑖 Δ𝑉S𝑖 Δ𝑊S𝑖 ] = ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄𝑖 𝐺𝑖 = [ , , 2 2 2 2. 0 𝑣Gx𝑖 0 1 [𝑣Gy𝑖 ] = [ 0 ] + [ 𝜔𝑧 2 −𝜔 𝑣Gz𝑖 𝑞̇ 𝑖 𝑦. 𝑣G𝑖. 𝑣Gx𝑖 𝑐𝑣11 1 = [𝑣Gy𝑖 ] = {[𝑐𝑣21 2 𝑐𝑣 𝑣Gz𝑖 31. 𝑐𝑣12 𝑐𝑣22 𝑐𝑣32. −𝜔𝑧 0 𝜔𝑥. 𝜔𝑦 Δ𝑈S𝑖 −𝜔𝑥 ] [ Δ𝑉S𝑖 ] 0 Δ𝑊S𝑖. 𝑐𝑣13 𝑞̇ 𝐴 0 0 𝑑𝑣13 0 𝑐𝑣23] [𝑞̇ 𝐵 ] + [0 0 𝑑𝑣23 ] [ 0 ]} 𝑐𝑣33 𝑞̇ 𝐶 0 0 𝑑𝑣33 𝑞̇ 𝑖. Δ𝑈S𝐴 Δ𝑊S𝐴 Δ𝑉S𝐴 𝑐𝑣12 = Δ𝑊S𝐴 𝑐𝑣13 = 1 1 Δ𝑈S𝐴 Δ𝑈S𝑖 Δ𝑈Sc Δ𝑊S𝑖 [ ] 𝑐21 = − + Δ𝑉S𝑖 Δ𝑊S𝐴 Δ𝑊S𝐶 1 Δ𝑉Sc Δ𝑊Si Δ𝑈Si Δ𝑉Sa [ ] 𝑐22 = − + Δ𝑉S𝑖 Δ𝑊Sc Δ𝑊S𝐴 1 [Δ𝑊Si + Δ𝑈Si ] 𝑐23 = − Δ𝑉S𝑖 𝑐𝑣11 =.
(130) 130. Δ𝑈S𝐶 Δ𝑊S𝐶 Δ𝑉S𝐶 𝑐32 = Δ𝑊S𝐶 𝑐33 = 1 𝑑𝑣13 = 0 Δ𝑊S𝑖 𝑑𝑣23 = Δ𝑉Si 𝑑𝑣33 = 1 𝑐31 =. 𝑎𝐻𝑖 = 𝑎𝑄𝑖 + 𝜔𝑖 ×𝜔𝑖 × 𝑄𝑖 𝑏𝐻𝑖 + 𝛼𝑖 × 𝑄𝑖𝑏𝐻𝑖. 𝑎𝐻𝑖 𝑎𝐻𝑖 = 𝑎𝑃 = [𝑢̈ , 𝑣̈ , 𝑤̈]𝑇 𝑎𝑄𝑖 𝑎𝑄𝑖 = [0,0, 𝑞̈ 𝑖 ]𝑇 𝑄𝑖. 𝑏𝐻𝑖. ̂ 𝑖 𝑄𝑖𝑏𝐻𝑖 = 𝜔𝑖 × 𝑄𝑖𝑏𝐻𝑖 𝝎 ̂ 𝑖 𝑄𝑖𝑏𝐻𝑖 = 𝛼𝑖 × 𝑄𝑖 𝑏𝐻𝑖 𝜶.
(131) 131. ̂𝑖 α 0 ̂ 𝑖 = [ 𝛼𝑧 𝜶 −𝛼𝑦. −𝛼𝑧 0 𝛼𝑥. 0 𝑎𝑁𝑥 0 𝑢̈ [ 𝑣̈ ] = [ 0 ] + [𝑎𝑁𝑦 ] + [ 𝛼𝑧 𝑎𝑁𝑧 𝑞𝑖̈ −𝛼𝑦 𝑤̈ 0 𝑢̈ − 𝑎𝑁𝑥 [ 𝑣̈ − 𝑎𝑁𝑦 ] = [ 𝛼𝑧 −𝛼𝑦 𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 ). 𝛼𝑦 −𝛼𝑥 ] 0. −𝛼𝑧 0 𝛼𝑥 −𝛼𝑧 0 𝛼𝑥. 𝛼𝑦 Δ𝑈S𝑖 −𝛼𝑥 ] [ Δ𝑉S𝑖 ] 0 Δ𝑊S𝑖 𝛼𝑦 Δ𝑈S𝑖 −𝛼𝑥 ] [ Δ𝑉S𝑖 ] 0 Δ𝑊S𝑖. 𝑎𝑁𝑥 𝑎𝑁𝑥 = 𝜔𝑦 (𝜔𝑥 Δ𝑉S𝑖 − 𝜔𝑦 Δ𝑈S𝑖 ) − 𝜔𝑧 (𝜔𝑧 Δ𝑈S𝑖 − 𝜔𝑥 Δ𝑊S𝑖 ) 𝑎𝑁𝑦 𝑎𝑁𝑦 = 𝜔𝑧 (𝜔𝑦 Δ𝑊S𝑖 − 𝜔𝑧 Δ𝑉S𝑖 ) − 𝜔𝑥 (𝜔𝑥 Δ𝑉S𝑖 − 𝜔𝑦 Δ𝑈S𝑖 ) 𝑎𝑁𝑧 𝑎𝑁𝑧 = 𝜔𝑥 (𝜔𝑧 Δ𝑈S𝑖 − 𝜔𝑥 Δ𝑊S𝑖 ) − 𝜔𝑦 (𝜔𝑦 Δ𝑊S𝑖 − 𝜔𝑧 Δ𝑉S𝑖 ). ̅̅̅̅̅̅ 𝑄𝑖𝐻𝑖. 𝛼𝑖 ∙ 𝑄𝑖 𝑏𝐻𝑖 = 0 𝛼𝑥 Δ𝑈S𝑖 + 𝛼𝑦 Δ𝑉S𝑖 + 𝛼𝑧 Δ𝑊S𝑖 = 0. 0 [Δ𝑈S𝑖 Δ𝑉S𝑖. Δ𝑊S𝑖 Δ𝑉S𝑖 −Δ𝑈S𝑖. −Δ𝑉S𝑖 ∝𝑥 𝑢̈ − 𝑎𝑁𝑥 Δ𝑊S𝑖 ] [∝𝑦 ] = [ 0 ] ∝𝑧 𝑤̈ − (𝑞̈ 𝑖 + 𝑎𝑁𝑧 ) 0.
(132) 132. 1 Δ𝑈S𝑖 2 Δ𝑈S𝑖 2 ) {( ) (𝑢̈ − 𝑎𝑁𝑥 ) + [1 − ( ) ] [𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 )]} 𝛼𝑥 = ( Δ𝑉S𝑖 𝑙 𝑙 𝛼𝑦 =. Δ𝑊S𝑖 Δ𝑈 (𝑢̈ − 𝑎𝑁𝑥 ) − 2S𝑖 [𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 )] 2 𝑙 𝑙. 1 Δ𝑊S𝑖 2 Δ𝑈S𝑖 ∙ Δ𝑊S𝑖 {[( ) − 1] (𝑢̈ − 𝑎𝑁𝑥 ) − ( ) [𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 )]} 𝛼𝑧 = Δ𝑉S𝑖 𝑙 𝑙2 Δ𝑈S𝑖 2 Δ𝑈S𝑖 2 ( ) (𝑢̈ − 𝑎𝑁𝑥 ) + [1 − ( ) ] [𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 )] 𝑙 𝑙 𝛼𝑥𝑖 1 Δ𝑉Si Δ𝑊S𝑖 Δ𝑉 Δ𝑈 𝛼 Α𝑖 = [ 𝑦𝑖 ] = (𝑢̈ − 𝑎𝑁𝑥 ) − Si 2 S𝑖 [𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 )] 2 Δ𝑉Si 𝑙 𝑙 𝛼𝑧𝑖 2 Δ𝑊S𝑖 Δ𝑈S𝑖 Δ𝑊S𝑖 [( ) − 1] (𝑢̈ − 𝑎𝑁𝑥 ) − ( ) [𝑤̈ − (𝑞𝑖̈ + 𝑎𝑁𝑧 )] 𝑙 𝑙2 [ ]. 𝑎𝐺𝑖 = 𝑎𝑄𝑖 + 𝜔𝑖 ×𝜔𝑖 × 𝑄𝑖𝑏𝐻𝑖 + 𝛼𝑖 × 𝑄𝑖𝑏𝐻𝑖. 0 𝑎Gx𝑖 0 [𝑎Gy𝑖 ] = [ 0 ] + [ 𝜔𝑧 𝑎Gz𝑖 𝑞̈ 𝑖 −𝜔𝑦. 𝑎𝐺𝑖. −𝜔𝑧 0 𝜔𝑥. 𝜔𝑦 𝑄𝑖𝒗𝐺𝑥𝑖 0 1 𝑄𝑖 −𝜔𝑥 ] [ 𝒗𝐺𝑦𝑖 ] + [ 𝛼𝑧 2 −𝛼 𝑄𝑖 0 𝑦 𝒗 𝐺𝑧𝑖. −𝛼𝑧 0 𝛼𝑥. 𝛼𝑦 Δ𝑈S𝑖 −𝛼𝑥 ] [ Δ𝑉S𝑖 ] 0 Δ𝑊S𝑖. 𝜔𝑦 ( 𝑄𝑖𝒗𝐺𝑧𝑖 )−𝜔𝑧 ( 𝑄𝑖𝒗𝐺𝑦𝑖 ) 𝛼𝑦 (Δ𝑊S𝑖 ) − 𝛼𝑧 (Δ𝑉S𝑖 ) 𝑎Gx𝑖 0 1 = [𝑎Gy𝑖 ] = [ 0 ] + [ 𝜔𝑧 ( 𝑄𝑖𝒗𝐺𝑥𝑖 )−𝜔𝑥 ( 𝑄𝑖𝒗𝐺𝑦𝑖 ) ] + [𝛼𝑧 (Δ𝑈S𝑖 ) − 𝛼𝑥 (Δ𝑊S𝑖 )] 2 𝑎Gz𝑖 𝑞̈ 𝑖 𝛼𝑥 (Δ𝑉S𝑖 ) − 𝛼𝑦 (Δ𝑈S𝑖 ) 𝜔 ( 𝑄𝑖 𝒗 )−𝜔 ( 𝑄𝑖𝒗 ) 𝑥. 𝐺𝑦𝑖. 𝑦. 𝐺𝑥𝑖.
(133) 133.
(134) 134.
(135) 135. 𝑖 𝐶 𝑛𝑖. =. 𝑑 𝑖 𝐶 ( ℎ𝑖 ) 𝑑𝑡. 𝑖 𝐶 𝑛𝑖 𝑖 𝐶 ℎ𝑖. 𝑖 ℎ𝐶 𝑖. = 𝑚𝑖 𝑒𝑖 ( 𝑖𝑠𝑖 × 𝑖 𝑣𝐺𝑖 ) + 𝑖 ℎ𝑖𝐺 + 2𝑚𝐽 𝑒𝑖 ( 𝑖𝑠𝑖 × 𝑖 𝑣𝐸𝑖 ).
(136) 136. 𝑚𝑖 𝑚𝐽 𝑒𝑖. 𝑒𝑖 =. 𝑙𝑖 2. 𝑙𝑖 𝑖𝑠 𝑖 𝑖 𝑖 𝑖. 𝑠𝑖 = [0, 0, 1]𝑇. 𝑉𝐺𝑖 𝑉𝐸𝑖. 𝑖 𝐶 ℎ𝑖. 𝑖 𝐺 ℎ𝑖. 𝑖𝑰. 𝑖. 𝑖 𝑖. 𝑖. = 𝑖 𝐼𝑖 𝑖𝜔𝑖. 𝐼𝑖𝑥𝑥. 𝑖. 𝐼𝑖𝑥𝑦. 𝑖. 𝐼𝑖𝑥𝑧. 𝑰𝑖 = [𝑖 𝐼𝑖𝑦𝑥 𝑖 𝐼𝑖𝑧𝑥. 𝑖. 𝐼𝑖𝑦𝑦. 𝑖. 𝐼𝑖𝑦𝑧 ]. 𝑖. 𝐼𝑖𝑧𝑦. 𝑖. 𝐼𝑖𝑧𝑧. 𝜔𝑖 𝑖. 𝜔𝑖 = ( 𝑖 𝜔𝑖𝑢´, 𝑖 𝜔𝑖𝑤´, 0). 𝑖. 𝑖. 𝐼𝑖𝑥𝑥 𝑰𝑖 = [ 0 0. 𝑖. 0 𝐼𝑖𝑦𝑦 0. 0 0 ] 𝑖 𝐼𝑖𝑧𝑧.
(137) 137. 𝑖. 𝐼𝑖𝑥𝑥 = 𝑖𝐼𝑖𝑦𝑦. 𝑑 𝑖 𝐶 ( ℎ𝑖 ) = 𝑚𝑖 𝑒𝑖 ( 𝑖 𝑠𝑖 × 𝑖 𝑎𝐺𝑖 ) + 2𝑚𝐽 𝑒𝑖 ( 𝑖𝑠𝑖 × 𝑖𝑎𝐸𝑖 ) + 𝑖 𝑰1𝑖 𝑖 𝛼𝑖 + 𝑖 𝜔𝑖 ×( 𝑖𝑰1𝑖 𝑖 𝜔𝑖 ) 𝑑𝑡. 𝑖. 𝑎𝐸𝑖 = 𝑖 𝑣̇ 𝐸𝑖. 𝑖. 𝑎𝐺𝑖 = 𝑖 𝑣̇ 𝐺𝑖 𝑖. 𝛼𝑖 = 𝑖 𝜔̇ 𝑖 𝑖. 𝜔̇ 𝑖 = 𝑖 𝛼𝑖 = ( 𝑖 𝛼𝑖𝑢´, 𝑖 𝛼𝑖𝑣´, 0). −𝑚𝑖 𝑒𝑖 𝑖 𝑎𝐺𝑖𝑣´ − 2𝑚𝐽 𝑒𝑖 𝑖 𝑎𝐸𝑖𝑣´ + 𝑖 𝐼𝑖𝑥𝑥 𝑖 𝛼𝑖𝑢´ 𝑑 𝑖 𝐶 ( ℎ𝑖 ) = [ 𝑚𝑖 𝑒𝑖 𝑖𝑎𝐺𝑖𝑢´ + 2𝑚𝐽 𝑒𝑖 𝑖𝑎𝐸𝑖𝑢´ + 𝑖𝐼𝑖𝑦𝑦 𝑖 𝛼𝑖𝑣´ ] 𝑑𝑡 0. 𝑂. 𝑖 𝑖. 𝑔 = [0, 0, −𝑔𝐺 ]𝑇. 𝑓𝐶𝑖 = [ 𝑖 𝑓𝐶𝑖𝑢´, 𝑖 𝑓𝐶𝑖𝑣´, 𝑖𝑓𝐶𝑖𝑤´]𝑇. 𝑓𝐸𝑖 = [ 𝑖 𝑓𝐸𝑖𝑢´, 𝑖 𝑓𝐸𝑖𝑣´ , 𝑖 𝑓𝐸𝑖𝑤´]𝑇.
(138) 138. 𝑖 𝐶 𝑛𝑖. = 2𝑒𝑖 𝑖 𝑠𝑖 ×(− 𝑖𝑓𝐸𝑖 ) + 𝑚𝑖 𝑒𝑖 𝑖𝑠𝑖 × 𝑖 𝑹𝑂 𝑂𝑔 + 𝑚𝐽 (2𝑒𝑖 )( 𝑖𝑠𝑖 × 𝑖𝑹𝑂 𝑂𝑔). 2𝑒𝑖 𝑖 𝑓𝐸𝑖𝑣´ 𝑖 𝐶 𝑛𝑖 = [−2𝑒𝑖 𝑖 𝑓𝐸𝑖𝑢´ + 𝑚𝑖 𝑒𝑖 𝑔𝐺 sin 𝜃𝑖 + 2𝑚𝐽 𝑒𝑖 𝑔𝐺 sin 𝜃𝑖 ] 0. 𝑖. 𝑓𝐸𝑖𝑥 = 𝑖. 𝑖. 𝑓𝐸𝑖𝑦 =. 𝑓𝐹𝑖𝑥 = 𝑖. 𝑚𝑖 𝑒𝑖 𝑔𝐺 sin 𝜃𝑖 + 2𝑚𝐽 𝑒𝑖 𝑔𝐺 sin 𝜃𝑖 − 1 [ 𝑖 ] 2𝑒𝑖 − 𝐼𝑖𝑦𝑦 𝑖𝛼𝑖𝑣´ − 𝑚𝑖 𝑒𝑖 𝑖 𝑎𝐺𝑖𝑢´ − 2𝑚𝐽 𝑒𝑖 𝑖 𝑎𝐸𝑖𝑢´ 1 𝑖 [ 𝐼𝑖𝑥𝑥 𝑖 𝛼𝑖𝑢´ − 𝑚𝑖 𝑒𝑖 𝑖 𝑎𝐺𝑖𝑣´ − 2𝑚𝐽 𝑒𝑖 𝑖 𝑎𝐸𝑖𝑣´ ] 2𝑒𝑖. 𝑚𝑖 𝑒𝑖 𝑔𝐺 sin 𝜃𝑖 + 2𝑚𝐽 𝑒𝑖 𝑔𝐺 sin 𝜃𝑖 − 1 [ 𝑖 ] 2𝑒𝑖 − 𝐼𝑖𝑦𝑦 𝑖 𝛼𝑖𝑣´ − 𝑚𝑖 𝑒𝑖 𝑖 𝑎𝐺𝑖𝑢´ − 2𝑚𝐽 𝑒𝑖 𝑖 𝑎𝐸𝑖𝑢´. 𝑓𝐹𝑖𝑦 =. 1 𝑖 [ 𝐼 𝑖 𝛼 − 𝑚𝑖 𝑒𝑖 𝑖 𝑎𝐺𝑖𝑣´ − 2𝑚𝐽 𝑒𝑖 𝑖 𝑎𝐸𝑖𝑣´ ] 2𝑒𝑖 𝑖𝑥𝑥 𝑖𝑢´.
(139) 139. 3. ∑( 𝑂𝑓𝐸𝑖 + 𝑂𝑓𝐹𝑖 ) − 𝑚𝑃 𝑂𝑔 = 𝑚𝑃 𝑂𝑎𝑝 𝑖=1. 𝑚𝑃 𝑂 𝑂. 𝑂. 𝑎𝑝. 𝑓𝐸𝑖 𝑂. 𝑓𝐸𝑖 = 𝑂𝑹𝑖 𝑖 𝑓𝐸𝑖. 𝑂. 𝑓𝐹𝑖 = 𝑂𝑹𝑖 𝑖 𝑓𝐹𝑖. 𝑓𝐹𝑖. 3. ∑( 𝑖=1. ( 𝑖 𝑓𝐸𝑖𝑥 + 𝑖𝑓𝐹𝑖𝑥 )𝑐𝜑𝑖 𝑐𝜃𝑖 − ( 𝑖𝑓𝐸𝑖𝑦 + 𝑖 𝑓𝐹𝑖𝑦 )𝑠𝜑𝑖 + + ( 𝑖 𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )𝑐𝜑𝑖 𝑠𝜃𝑖. ) = 𝑚𝑝𝑣̇𝑝𝑥. 3. ( 𝑖 𝑓𝐸𝑖𝑥 + 𝑖 𝑓𝐹𝑖𝑥 )𝑠𝜑𝑖 𝑐𝜃𝑖 − ( 𝑖 𝑓𝐸𝑖𝑦 + 𝑖𝑓𝐹𝑖𝑦 )𝑐𝜑𝑖 + ∑( ) = 𝑚𝑝𝑣̇𝑝𝑦 + ( 𝑖 𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )𝑠𝜑𝑖 𝑠𝜃𝑖 𝑖=1. 3. ∑(−( 𝑖 𝑓𝐸𝑖𝑥 + 𝑖 𝑓𝐹𝑖𝑥 )𝑠𝜃𝑖 + ( 𝑖 𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )𝑐𝜑𝑖 ) = 𝑚𝑝𝑣̇𝑝𝑧 + 𝑚𝑝 𝑔𝑐 𝑖=1.
(140) 140. 3 𝑃. 𝑛5 = ∑[ 𝑃𝑏𝐸𝑖 ×( 𝑃𝑓𝐸𝑖 ) + 𝑃𝑏𝐹𝑖 ×( 𝑃𝑓𝐹𝑖 )] 𝑖=1. 𝑃. 𝑛5 = 𝑃𝑰𝐺5 𝑃𝛼𝐺5 + 𝑃𝜔𝐺5 ×( 𝑃𝑰𝐺5 𝑃𝜔𝐺5 ). 𝑃. 𝑛5 = 0. 3. ∑[ 𝑃𝑏𝐸𝑖 ×( 𝑃𝑓𝐸𝑖 ) + 𝑃𝑏𝐹𝑖 ×( 𝑃𝑓𝐹𝑖 )] = 0 𝑖=1. 𝑃. 𝑏𝐸𝑖. 𝑃. 𝑃. 𝑏𝐸𝑖 = [ 𝑏𝐸𝑖𝑢 , 𝑏𝐸𝑖𝑣 , 𝑏𝐸𝑖𝑤 ]𝑇. 𝑃. 𝑏𝐹𝑖 = [ 𝑏𝐹𝑖𝑢 , 𝑏𝐹𝑖𝑣 , 𝑏𝐹𝑖𝑤 ]𝑇. 𝑏𝐹𝑖. 𝑃. 𝑃. 𝑓𝐸𝑖 𝑃. 𝑓𝐸𝑖 = 𝑃𝑅𝑂 𝑂𝑓𝐸𝑖 = 𝑃𝑅𝑖 𝑖 𝑓𝐸𝑖. 𝑃. 𝑓𝐹𝑖 = 𝑃𝑅𝑂 𝑂 𝑓𝐹𝑖 = 𝑃𝑅𝑖 𝑖 𝑓𝐹𝑖. 𝑓𝐹𝑖. 3. ∑ 𝑏𝑖𝑣 [𝑘31 ( 𝑖 𝑓𝐸𝑖𝑥 + 𝑖 𝑓𝐹𝑖𝑥 ) + 𝑘32 ( 𝑖𝑓𝐸𝑖𝑦 + 𝑖 𝑓𝐹𝑖𝑦 ) + 𝑘33 ( 𝑖𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )] = 0 𝑖=1.
(141) 141. 3. ∑ − 𝑏𝑖𝑢 [𝑘31 ( 𝑖𝑓𝐸𝑖𝑥 + 𝑖 𝑓𝐹𝑖𝑥 ) + 𝑘32 ( 𝑖𝑓𝐸𝑖𝑦 + 𝑖 𝑓𝐹𝑖𝑦 ) + 𝑘33 ( 𝑖𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )] = 0 𝑖=1 3. ∑{ 𝑏𝑖𝑢 [𝑘21 ( 𝑖 𝑓𝐸𝑖𝑥 + 𝑖𝑓𝐹𝑖𝑥 ) + 𝑘22 ( 𝑖𝑓𝐸𝑖𝑦 + 𝑖 𝑓𝐹𝑖𝑦 ) + 𝑘23 ( 𝑖𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )] 𝑖=1. + 𝑏𝑖𝑣 [𝑘11 ( 𝑖 𝑓𝐸𝑖𝑥 + 𝑖 𝑓𝐹𝑖𝑥 ) + 𝑘12( 𝑖 𝑓𝐸𝑖𝑦 + 𝑖 𝑓𝐹𝑖𝑦 ) + 𝑘13 ( 𝑖𝑓𝐸𝑖𝑧 + 𝑖 𝑓𝐹𝑖𝑧 )]}. 𝑘𝑖𝑗 𝑃. 𝑖 𝑖. 𝑃. 𝑹 𝑖 = 𝑹𝑂. 𝑂. 𝑘11 𝑹𝑖 = [𝑘21 𝑘31. 𝑘12 𝑘22 𝑘32. 𝑘13 𝑘23 ] 𝑘33. 𝑓𝐶𝑖 + 𝑖𝑓𝐸𝑖 + (𝑚𝑖 + 2𝑚𝑗 ) 𝑖 𝑹𝑂 𝑂𝑔 = 𝑚𝑖 𝑖 𝑎𝐺𝑖 + 𝑚𝑗 𝑖 𝑎𝐸𝑖 + 𝑚𝑗 𝑖𝑎𝐶𝑖. 𝑓𝐷𝑖 + 𝑖 𝑓𝐹𝑖 + (𝑚𝑖 + 2𝑚𝑗 ) 𝑖 𝑹𝑂 𝑂𝑔 = 𝑚𝑖 𝑖 𝑎𝐺𝑖 + 𝑚𝑗 𝑖 𝑎𝐹𝑖 + 𝑚𝑗 𝑖 𝑎𝐷𝑖.
(142) 142. 𝐶. 𝑓𝐴𝑖 − 𝐶 𝑓𝐶𝑖 − 𝐶 𝑓𝐷𝑖 + 𝑚1𝑖 𝑂𝑔 = 𝑚1𝑖 𝑎1𝑖. 𝑚1𝑖. 𝑎1𝑖 𝐶. 𝑓𝐴𝑖 𝐶. 𝐶. 𝑓𝐴𝑖 = 𝐶 𝑹𝑖 𝑖𝑓𝐴𝑖 = [ 𝐶 𝑓𝐴𝑖𝑥 , 𝐶 𝑓𝐴𝑖𝑦 , 𝐹𝑖 ]𝑇. 𝑓𝐶𝑖 𝐶. 𝑓𝐶𝑖 = 𝐶 𝑹𝑖 𝑖 𝑓𝐶𝑖 = [ 𝐶 𝑓𝐶𝑖𝑥 , 𝐶 𝑓𝐶𝑖𝑦 , 𝐶 𝑓𝐶𝑖𝑧 ]𝑇.
(143) 143. 𝐶. 𝑓𝐷𝑖 𝐶. 𝑓𝐷𝑖 = 𝐶 𝑹𝑖 𝑖 𝑓𝐷𝑖 = [ 𝐶 𝑓𝐷𝑖𝑥 , 𝐶 𝑓𝐷𝑖𝑦 , 𝐶 𝑓𝐷𝑖𝑧 ]𝑇. 𝐶 𝑓𝐷𝑖𝑥 + 𝐶 𝑓𝐶𝑖𝑥 𝑓𝐴𝑖𝑥 𝐶 𝑓𝐷𝑖𝑦 + 𝐶 𝑓𝐶𝑖𝑦 [ 𝐶 𝑓𝐴𝑖𝑦 ] = [ ] 𝐶 𝐹𝑖 𝑓𝐷𝑖𝑧 + 𝐶 𝑓𝐶𝑖𝑧 + 𝑚1𝑖 𝑔𝐺 + 𝑚1𝑖 𝑞̈ 𝑖 𝐶. 3 𝐴. 𝑛1𝑖 = ∑[ 𝐴𝑏𝐶𝑖 ×(− 𝐶 𝑓𝐶𝑖 ) + 𝐴𝑏𝐷𝑖 ×(− 𝐶 𝑓𝐷𝑖 )] + 𝑛𝐴𝑖 𝑖=1 𝐴. 𝐴. 𝑛1𝑖 = 𝑃𝑰𝐺1𝑖 𝑃𝛼𝐺1𝑖 + 𝑃𝜔𝐺1𝑖 ×( 𝑃𝑰𝐺1𝑖 𝑃𝜔𝐺1𝑖 ). 𝑏𝐶𝑖 𝐴. 𝐴. 𝑏𝐶𝑖 = [𝑏𝐶𝑖𝑥 , 𝑏𝐶𝑖𝑦 , 𝑏𝐶𝑖𝑧 ]𝑇. 𝑏𝐶𝑖 𝐴. 𝑏𝐷𝑖 = [𝑏𝐷𝑖𝑥 , 𝑏𝐷𝑖𝑦 , 𝑏𝐷𝑖𝑧 ]𝑇. 3. 𝑛𝐴𝑖 = ∑[ 𝐴𝑏𝐶𝑖 ×( 𝐶 𝑓𝐶𝑖 ) + 𝐴𝑏𝐷𝑖 ×( 𝐶 𝑓𝐷𝑖 )] 𝑖=1.
(144) 144. 𝐾. 𝜕γ𝑗 𝑑 𝜕𝐿 𝜕𝐿 𝜕𝐷p ( )− + − ∑ 𝜆𝑗 = 𝑄𝑘 𝑑𝑡 𝜕𝑞̇ 𝑘 𝜕𝑞𝑘 𝜕𝑞̇ 𝑘 𝜕𝑞𝑘 𝑖=1. 𝑘 𝑗 𝐾. 𝑛 𝐿 𝐿= 𝑇−𝑉 𝑇 𝑉 𝐷P. 𝑞𝑖 𝜆𝑗 Γ𝑗 𝑄𝑖. 𝜕𝐷P =0 𝜕𝑞̇ 𝑖.
(145) 145. 𝑞A 𝑞B 𝑞C. Γ𝑗. 𝑄𝑗. 𝐻𝑗. 2. 𝛤𝑗 = ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄𝑗 𝐻𝑗 − ⃗𝑙𝑗. 2. 2. 2 2 2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑄𝑗 𝐻𝑗 = (𝑢 + ℎ𝑥𝑗 − 𝑥𝐴𝑗 ) + (𝑣 + ℎ𝑦𝑗 − 𝑦𝐴𝑗 ) + (𝑤 − 𝑞𝑗 ) ⃗⃗⃗⃗ 𝑙2=𝑙2 2. 𝑗. 𝑗. 2. 2. 𝛤𝑗 = (𝑢 + ℎ𝑥𝑗 − 𝑥𝐴𝑗 ) + (𝑣 + ℎ𝑦𝑗 − 𝑦𝐴𝑗 ) + (𝑤 − 𝑞𝑗 ) − 𝑙𝑗 2. 3. 𝑑 𝜕𝐿 𝜕𝐿 𝜕Γ𝑖 ( )− − ∑ 𝜆𝑖 = 𝑓𝑢 𝑑𝑡 𝜕𝑢̇ 𝜕𝑢 𝜕𝑢 𝑖=1 3. 𝑑 𝜕𝐿 𝜕𝐿 𝜕Γ𝑖 ( )− − ∑ 𝜆𝑖 = 𝑓𝑣 𝑑𝑡 𝜕𝑣̇ 𝜕𝑣 𝜕𝑣 𝑖=1 3. 𝑑 𝜕𝐿 𝜕𝐿 𝜕Γ𝑖 ( )− − ∑ 𝜆𝑖 = 𝑓𝑤 𝑑𝑡 𝜕𝑤̇ 𝜕𝑤 𝜕𝑤 𝑖=1.
(146) 146. 𝑞A 𝑞B 𝑞C 3. 𝑑 𝜕𝐿 𝜕𝐿 𝜕Γ𝑖 ( )− − ∑ 𝜆𝑖 = 𝐹𝐴 𝑑𝑡 𝜕𝑞̇ 𝐴 𝜕𝑞𝐴 𝜕𝑞𝐴 𝑖=1 3. 𝑑 𝜕𝐿 𝜕𝐿 𝜕Γ𝑖 ( )− − ∑ 𝜆𝑖 = 𝐹𝐵 𝑑𝑡 𝜕𝑞̇ 𝐵 𝜕𝑞𝐵 𝜕𝑞𝐵 𝑖=1 3. 𝑑 𝜕𝐿 𝜕𝐿 𝜕Γ𝑖 ( )− − ∑ 𝜆𝑖 = 𝐹𝐶 𝑑𝑡 𝜕𝑞̇ 𝐶 𝜕𝑞𝐶 𝜕𝑞𝐶 𝑖=1. 𝐿= 𝑇−𝑉. 3. 3. 3. 𝑇 = 𝑇𝑝 + ∑(𝑇𝑞𝑘 + 𝑇𝑙𝑘 ) = 𝑇𝑃 + ∑ 𝑇𝑞𝑘 + ∑ 𝑇𝑙𝑘 𝑘=1. 𝑘=1. 𝑇𝑝 𝑇𝑞𝑘 𝑇𝑙𝑘 𝑇𝑙𝑘 = 𝑇3𝑘 + 𝑇4𝑘 𝑇3𝑘 𝑇4𝑘. 𝑘=1.
(147) 147. 1 1 𝑇𝑝 = 𝑚𝑝 𝑣𝑝 𝑇 𝑣𝑝 = 𝑚𝑝 (𝑢̇ 2 +𝑣̇ 2 + 𝑤̇ 2 ) 2 2 𝑞A 𝑞B 𝑞C. 𝑇𝑝 =. 1 𝑇 𝑞̇ (𝑚𝑝 ∙ 𝐽𝑇 𝐽)𝑞̇ 2. 3. 3. 𝑘=1. 𝑘=1. 1 1 𝑇𝑞 = 𝑚1 ∙ 𝑞̇ 𝑇 ∙ 𝑞̇ = ∑ 𝑇𝑞𝑘 = ∑ 𝑚1𝑘 ∙ 𝑞̇ 𝑘 2 2 2. 1 1 𝑇I = 𝑚I 𝑣G𝑖 T 𝑣𝐺𝑖 + 𝜔𝑖 𝑇 𝐼𝐺𝑖 𝜔𝑖 2 2. 𝑇𝑖 𝑣𝐺𝑖 𝜔𝑖 𝑚𝑖 𝐼𝐺𝑖. ω ω. ω.
(148) 148. 𝑞A. 𝑞B. 𝑞C. 1 𝑇𝑖 = 𝑞̇ 𝑇 ∙ (𝑚𝑖 ∙ 𝐽𝑣𝑖 𝑇 ∙ 𝐽𝑣𝑖 + 𝐽𝜔𝑖 𝑇 ∙ 𝐼𝐺𝑖 ∙ 𝐽𝜔𝑖 ) ∙ 𝑞̇ 2. 6. 6. 𝑘=1. 𝑘=1. 1 𝑇𝑙 = ∑ 𝑇𝑙𝑘 = 𝑞̇ 𝑇 ∙ ∑(𝑚𝑘 ∙ 𝐽𝑣𝑘 𝑇 ∙ 𝐽𝑣𝑘 + 𝐽𝜔𝑘 𝑇 ∙ 𝐼𝐺𝑘 ∙ 𝐽𝜔𝑘 ) ∙ 𝑞̇ 2. 6. 1 𝑇(𝑞, 𝑞̇ ) = 𝑞̇ 𝑇 [𝑚1 𝐼 + 𝑚𝑝 𝐽𝑇 𝐽 + ∑(𝑚𝑘 𝐽𝑣𝑘 𝑇 𝐽𝑣𝑘 + 𝐽𝜔𝑘 𝑇 𝐼𝐺𝑘 𝐽𝜔𝑘 )] 𝑞̇ 2 𝑘=1. 𝑇=. 1 𝑇 𝑞̇ 𝑩(𝒒)𝑞̇ 2. 𝑇 𝑞, 𝑝, 𝑞̇ 𝑩(𝒒). 3. 3. 1 1 𝑇(𝑞, 𝑞̇ ) = 𝑞̇ 𝑇 𝑩(𝒒)𝑞̇ = ∑ ∑ 𝑏Ij (𝑞) 𝑞̇ 𝑖 𝑞̇ 𝑗 2 2 𝑖=1 𝑗=1. −𝑧.
(149) 149. 3. 𝑉 = 𝑉𝑝 + ∑(𝑉𝑞𝑘 + 𝑉𝑙𝑘 ) 𝑘=1 3. 3. 𝑉 = 𝑉𝑃 + ∑ 𝑉𝑞𝑘 + ∑ 𝑉𝑙𝑘 𝑘=1. 𝑘=1. 𝑉𝑝 𝑉𝑞𝑘 𝑉𝑙𝑘 𝑉𝑙𝑘 = 𝑉3𝑘 + 𝑉4𝑘 𝑉3𝑘 𝑉4𝑘. 𝑉𝑝 = 𝑚𝑝 g𝑤. 𝑙𝑐 𝑉𝑞𝑘 = 𝑚1𝑘 g [𝑞𝑘 − ] 2. 3. 3. 𝑉𝑙 = ∑ 𝑉𝑙𝑘 = ∑ 𝑚𝑘 g [ 𝑘=1. 𝑘=1. 𝑤 + 𝑞𝑘 ] 2.
(150) 150. 3. 3. 𝑘=1. 𝑘=1. 𝑙𝑐 𝑤 + 𝑞𝑘 ] 𝑉(𝑞, 𝑝) = 𝑚𝑝 g𝑤 + ∑ 𝑚1𝑘 g [𝑞𝑘 − ] + ∑ 𝑚𝑘 𝑔 [ 2 2. 𝑂. 𝑟𝑘 𝑁. 𝑉(𝑞, 𝑝) = ∑ 𝑚𝑘 g 𝑇 𝑂𝑟𝑘 𝑘=1. 𝐿(𝑞, 𝑞̇ ) = 𝑇(𝑞, 𝑞̇ ) − 𝑉(𝑞) 3. 3. 𝑁. 1 𝐿(𝑞, 𝑞̇ ) = ∑ ∑ 𝑏Ij (𝑞) 𝑞̇ 𝑖 𝑞̇ 𝑗 − ∑ 𝑚𝑘 g 𝑇 𝑂𝑟𝑘 2 𝑖=1 𝑗=1. 𝑘=1. 3. 𝜕𝐿 1 = ∑ 𝑞̇ 𝑗 (𝑏Kj + 𝑏Jk ) 𝜕𝑞̇ 𝑘 2 𝑗=1. 3. 3. 3. 𝜕𝑏Kj 𝑑 𝜕𝐿 1 1 ( ) = ∑ 𝑞J̈ (𝑏Kj + 𝑏Jk ) + ∑ ∑ 𝑞̇ 𝑞̇ 𝑑𝑡 𝜕𝑞̇ 𝑘 2 2 𝜕𝑞𝑖 𝑖 𝑗 𝑗=1. 𝑖=1 𝑗=1. 3. 3. 𝜕𝑏Ij 𝜕𝐿 1 𝜕𝑉(𝑞, 𝑝) = ∑∑ 𝑞̇ 𝑖 𝑞̇ 𝑗 − 𝜕𝑞𝑖 2 𝜕𝑞𝑘 𝜕𝑞𝑘 𝑖=1 𝑗=1.
(151) 151. 3. 3. 3. 𝐾. 𝜕𝑏Kj 𝜕𝑏Ij 𝜕Γ𝑗 1 1 𝜕𝑉(𝑞) ∑(𝑏Kj + 𝑏Jk )𝑞J̈ + ∑ ∑ ( − ) 𝑞̇ 𝑖 𝑞̇ 𝑗 + − ∑ 𝜆𝑗 = 𝑄𝑘 2 2 𝜕𝑞𝑖 𝜕𝑞𝑘 𝜕𝑞𝑘 𝜕𝑞𝑘 𝑗=1. 𝑖=1 𝑗=1. 3. 3. 𝑖=1. 3. 𝐾. 𝜕Γ𝑗 1 1 𝜕𝑉(𝑞) ∑(𝑏Kj + 𝑏Jk )𝑞J̈ + ∑ ∑ 𝑐𝑘𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗 + − ∑ 𝜆𝑗 = 𝑄𝑘 2 2 𝜕𝑞𝑘 𝜕𝑞𝑘 𝑗=1. 𝑐𝑘𝑖𝑗. 𝑖=1 𝑗=1. 𝑖=1. 𝜕𝑏Kj 𝜕𝑏Ij 𝑐𝑘𝑖𝑗 = ( − ) 𝜕𝑞𝑖 𝜕𝑞𝑘. 𝑏Kj + 𝑏Jk 𝜕𝑉(𝑞) 𝜕𝑞𝑘 𝜆𝑗. 𝜕Γ𝑗 𝜕𝑞𝑘. Φ(𝑞, 𝑞̇ , 𝑞̈ ) = 𝑩(𝒒)𝑞̈ + 𝑪(𝒒, 𝒒̇ ) + 𝐺 (𝑞) + 𝚪(𝒒, 𝝀) = 𝑄𝑘. 𝐵 (𝑞 ) 𝐶 (𝑞, 𝑞̇ ) 3. 3. 1 𝐶 (𝑞, 𝑞̇ ) = ∑ ∑ 𝑐𝑘𝑖𝑗 𝑞̇ 𝑖 𝑞̇ 𝑗 = 𝑞̇ 𝑇 𝐶𝑘 (𝑞)𝑞̇ 2 2 1. 𝐺 (𝑞 ) Γ(𝑞, 𝜆) 𝑄𝑘. 𝑖=1 𝑗=1.
(152) 152.
(153) 153. 𝑖. 𝑰𝑖 = [. 𝑖. 222,330 0 0 222,330 0 0. 5,297 1,909 𝑰𝑖 = [1,909 11,109 0 0. 0 0 ] [kgmm2 ] 0,211. 0 0 ] [kgmm2 ] 14,407. 𝑟0.
(154) 154. 𝑥𝑐 = 𝑦𝑐 = 0 𝑥 (𝑡) = 𝑟0 sin(𝑡) + 𝑥𝑐 𝑦(𝑡) = 𝑟0 cos(𝑡) + 𝑦𝑐 𝑧(𝑡) = 𝑡 + 200 mm.
(155) 155.
(156) 156.
(157) 157.
(158) 158.
(159) 159.
(160) 160.
(161) 161.
(162) 162.
(163) 163.
(164) 164.
(165) 165. 𝑞A 𝑞B 𝑞C. 𝑞A (𝑡) 𝑞B (𝑡) 𝑞C (𝑡). 𝑞A (𝑡) 𝑞B (𝑡) 𝑞C (𝑡). 𝑞A (𝑡) 𝑞B(𝑡) 𝑞C (𝑡).
(166) 166.
(167) 167. 150. 100. x [mm]. 50. 0 0. 50. 100. 150. -50. -100. -150. y [mm]. 200. 250. 300.
(168) 168. 150 100. x [mm]. 50 0 0. 50. 100. 150. -50 -100 -150. y [mm]. 200. 250. 300.
(169) 169. 150. 100. x [mm]. 50. 0 0. 50. 100. 150. -50. -100. -150. y [mm]. 200. 250. 300.
(170) 170. 𝑞(𝑡) = 𝑎0 𝑝0 (𝑡) + 𝑎1 𝑝1 (𝑡) + ⋯ + 𝑎𝑚 𝑝𝑚 (𝑡). 𝑝0 (𝑡), 𝑝1 (𝑡), … , 𝑝𝑚 (𝑡). 𝑎0 , 𝑎1 , … , 𝑎𝑚. 𝑛. 𝛾𝑗𝑖 = ∑ 𝑝𝑗 (𝑡𝑘 )𝑝𝑖 (𝑡𝑘 ) = 0 , 𝑘=0 𝑛. 𝛾𝑖𝑖 = ∑ 𝑝𝑗 (𝑡𝑘 )2 ≠ 0 𝑘=0. ∀𝑗 , 𝑖, 𝑗 ≠ 𝑖.
(171) 171. 𝑝0 (𝑡), 𝑝1 (𝑡), … , 𝑝𝑚 (𝑡) 𝑝𝑗 (𝑡) = (𝑡 − 𝛼𝑗 )𝑝𝑗−1 (𝑡) − 𝛽𝑗−1 𝑝𝑗−2 (𝑡), 𝛼𝑗. 𝑗 = 1, … , 𝑚 𝑝0 (𝑡) = 1. 𝛽𝑗−1 𝑝0 (𝑡) = 1 𝑝1 (𝑡) = 𝑡𝑝0 (𝑡) − 𝛼1 𝑝0 (𝑡) 𝑝2 (𝑡) = 𝑡𝑝1 (𝑡) − 𝛼2 𝑝1 (𝑡) − 𝛽1 𝑝0 (𝑡) ⋮ 𝑝𝑗 (𝑡) = 𝑡𝑝𝑗−1 (𝑡) − 𝛼𝑗 𝑝𝑗−1 (𝑡) − 𝛽𝑗−1 𝑝𝑗−2 (𝑡) 𝛼𝑗. 𝛽𝑗−1 2. 𝛼𝑗 =. ∑𝑛𝑘=0 𝑡𝑘 [𝑝𝑗−1 (𝑡𝑘 )] 2. ∑𝑛𝑘=0[𝑝𝑗−1 (𝑡𝑘 )]. 2. 𝛽𝑗 =. ∑𝑛𝑘=0 𝑡𝑘 [𝑝𝑗−1 (𝑡𝑘 )] 2. ∑𝑛𝑘=0[𝑝𝑗−1 (𝑡𝑘 )]. 𝑎0 , 𝑎1 , … , 𝑎𝑚 𝑎𝑗 =. ∑𝑛𝑘=0 𝑞𝑘 𝑝𝑗 (𝑡𝑘 ) 2. ∑𝑛𝑘=0[𝑝𝑗 (𝑡𝑘 )]. 𝑝0 (𝑡) = 1.
(172) 172. 𝜏 𝜏=. 𝑡 − 𝑡𝑎 , 𝑡𝑏 − 𝑡𝑎. 𝜏 ∈ [0,1]. 𝑞̃(𝑡) = 𝑞(𝑡) − 𝑞𝑎 (𝑞 − 𝜏) − 𝑞𝑏 𝜏.. 𝑡 = 𝑡𝑎 𝜏 = 0 𝑞(𝑡𝑎 ) = 𝑞𝑎 + 𝑞̃(0). 𝑞̃(0) = 𝑞(𝑡𝑎 ) − 𝑞𝑎. 𝑡 = 𝑡𝑏 𝜏 = 1 𝑞(𝑡𝑏 ) = 𝑞𝑏 + 𝑞̃ (1). 𝑞(𝑡𝑎 ). 𝑞̃(1) = 𝑞 (𝑡𝑏 ) − 𝑞𝑏. 𝑞(𝑡𝑏 ) 𝑞̃ = 0 𝑡 = 𝑡𝑎 𝑝0 (𝜏) = 𝜏(1 − 𝜏). 𝑡 = 𝑡𝑏. 𝑞̃(𝜏) = 𝑎0 𝑝0 (𝜏) + 𝑎1 𝑝1 (𝜏) + ⋯ + 𝑎𝑚 𝑝𝑚 (𝜏). 𝑝0 (𝜏) = 𝜏(1 − 𝜏) 𝑝1 (𝜏) = 𝜏𝑝0 (𝜏) − 𝛼1 𝑝0 (𝜏) 𝑝2 (𝜏) = 𝜏𝑝1 (𝜏) − 𝛼2 𝑝1 (𝜏) − 𝛽1 𝑝0 (𝜏) ⋮ 𝑝𝑗 (𝜏) = 𝜏𝑝𝑚−1 (𝜏) − 𝛼𝑚 𝑝𝑚−1 (𝜏) − 𝛽𝑚−1 𝑝𝑚−2 (𝜏) 𝑝0 (𝜏) = 𝜏 2 (1 − 𝜏) 𝑞̃(𝜏) = 𝑞(𝑡) − 𝑞𝑎 (1 − 𝜏 2 ) − 𝑞𝑏 𝜏 2.
(173) 173. 𝑝0 (𝜏) = 𝜏 3 (1 − 𝜏) 𝑞̃(𝜏) = 𝑞(𝑡) − 𝑞𝑎 (1 − 𝜏 3 ) − 𝑞𝑏 𝜏 3. 𝑚. 𝑚. 𝑘=1. 𝑘=1. 2𝜋𝑡 2𝜋𝑡 ) + ∑ 𝑏𝑘 sin (𝑘 ) 𝑞(𝑡) = 𝑎0 + ∑ 𝑎𝑘 cos (𝑘 𝑇 𝑇. 𝑛 𝑎0 𝑎𝑘 𝑦 𝑏𝑘 𝑇 𝑚 𝑛 = 2𝑚 + 1. 𝑎𝑘 𝑞0 1 𝑐1(𝑡0 ) 𝑞1 1 𝑐1 (𝑡1) ⋮ = ⋮ ⋮ 𝑞𝑛−2 ( 𝑐 𝑡 1 1 𝑛−2 ) [𝑞𝑛−1 ] [1 𝑐1 (𝑡𝑛−1 ). 𝑠1(𝑡0 ) 𝑠1(𝑡1 ) ⋮ 𝑠1(𝑡𝑛−2 ) 𝑠1(𝑡𝑛−1 ). 𝑏𝑘. ⋯ 𝑐𝑚 (𝑡0) ⋯ 𝑐𝑚(𝑡1 ) ⋮ ⋮ ( 𝑐 𝑡 ⋯ 𝑚 𝑛−2 ) ⋯ 𝑐𝑚 (𝑡𝑛−1 ). 𝑐𝑘 (𝑡) = cos (𝑘. 2𝜋𝑡 ) 𝑇. 𝑠𝑘 (𝑡) = sin (𝑘. 2𝜋𝑡 ) 𝑇. 𝑎0 𝑠𝑚 (𝑡0 ) 𝑎1 𝑠𝑚 (𝑡1 ) 𝑏1 ⋮ ⋮ 𝑠𝑚 (𝑡𝑛−2 ) 𝑎𝑚 𝑠𝑚 (𝑡𝑛−2 )] [ 𝑏𝑚 ].
(174) 174. 𝑛. 𝑛. 𝑞(𝑡) = ∑ (𝑞𝑘 ∏ 𝑘=0. 𝜋 sin (𝑇 (𝑡 − 𝑡𝑗 )). 𝜋 𝑗=0,𝑗≠𝑘 sin ( (𝑡𝑘 − 𝑡𝑗 )) 𝑇. ).
(175) 175. |𝑞̇ 𝑘 | ≤ 𝑞̇ máx |𝑢𝑘 | = |𝑞̈ 𝑘 | ≤ 𝑞̈ máx = 𝑈 𝑞̇ 𝑘 𝑞̈ 𝑘. 𝑟̇𝑘 𝑞𝑘. 𝑞̇ 𝑘.
(176) 176. 𝑞𝑘 ,. 𝑞̇ 𝑘 = 𝑞̇ 𝑘−1 + 𝑇𝑆 𝑢𝑘−1 𝑞𝑘 = 𝑞𝑘−1 +. 𝑇𝑆 (𝑞̇ + 𝑞̇ 𝑘+1 ) 2 𝑘 𝑞𝑘. 𝑒𝑘 =. 𝑞𝑘 − 𝑟𝑘 𝑈. 𝑒̇𝑘 =. 𝑞̇ 𝑘 − 𝑟̇𝑘 𝑈. 𝑧𝑘 =. 1 𝑒𝑘 𝑒̇𝑘 ( + ) 𝑇𝑆 𝑇𝑆 2 𝑧̇𝑘 =. 𝑒̇𝑘 𝑇𝑆.
(177) 177. 𝑚 = floor ( 𝜎𝑘 = 𝑧̇𝑘 +. 1 + √1 + 8|𝑧𝑘 | ) 2. 𝑧𝑘 𝑚 − 1 [sign(𝑧𝑘 )] + 𝑚 2. 1 + sign(𝑞̇ máx − 𝑈 ∙ 𝑇𝑆 + 𝑞̇ 𝑘 ∙ sign(𝜎𝑘 )) ] 𝑢𝑘 = [−𝑈 ∙ sat(𝜎𝑘 )] [ 2. −1 , 𝑥 < −1 −1 ≤ 𝑥 ≤ 1 sat(𝑥 ) = {𝑥 , +1 , 𝑥>1.
(178) 178.
(179) 179.
(180) 180.
(181) 181. 𝑃𝑥, 𝑃𝑦. 𝑃𝑧.
(182) 182.
(183) 183. 𝑥 (𝑡) = 𝑟𝑂 sin(𝑡) + 𝑥𝐶 𝑦(𝑡) = 𝑟𝑂 cos(𝑡) + 𝑦𝑐 𝑧 = 𝑧𝑂 𝑥 (𝑡) = 𝑟𝑂 sin(𝑡) + 𝑥𝐶 𝑦(𝑡) = 𝑟𝑂 cos(𝑡) + 𝑦𝑐 𝑧 = 𝑧𝑂 + 𝑡.
(184) 184.
(185) 185.
(186) 186.
(187) 187.
(188) 188.
(189) 189.
(190) 190.
(191) 191.
(192) 192.
(193) 193. •. •.
(194) 194.
(195) 195.
(196) 196. •. •. •.
(197) 197.
(198) 198. •.
(199) 199. •. •. •. •. •. •.
(200) 200. •. •. •. •. •. •. •. •.
(201) 201.
(202) 202.
(203) 203.
(204) 204.
(205) 205.
(206) 206.
(207) 207.
(208) 208.
(209) 209.
(210) 210.
(211) 211. 3. 1 1 𝑇 = (𝑚𝑖 + 𝑚𝑝 )(𝑢̇ 2 +𝑣̇ 2 + 𝑤̇ 2 ) + ∑(𝑚1𝑘 + 𝑚𝑖 ) ∙ 𝑞̇ 𝑘 2 2 2 𝑘=1 1 2 2 + 𝑚𝑖 (𝑣𝐺𝑖𝑥 + 𝑣𝐺𝑖𝑦 + 𝑣𝐺𝑖𝑧 2 ) 2 1 + [𝐼𝑥𝑥 𝜔𝑥𝑖 2 + 𝐼𝑦𝑦 𝜔𝑦𝑖 2 + 𝐼𝑧𝑧 𝜔𝑧𝑖 2 ] 2 𝑔𝑚1𝑘 𝑙1 𝑉 = 𝑔 𝑤(𝑚𝑝 + 𝑚𝑖 ) + 𝑔 𝑞𝑖 (𝑚1𝑘 + 𝑚𝑝 ) − 2.
(212) 212. 𝐹𝐴 = (𝑚1𝐴 + 𝑚𝐴 )𝑞𝐴̈ + 𝑔 (𝑚1𝐴 + 𝑚𝑝 ) − 2(𝑞𝐴 − 𝑤)𝜆1 𝐹𝐵 = (𝑚1𝐵 + 𝑚𝐵 )𝑞𝐵̈ + 𝑔 (𝑚1𝐵 + 𝑚𝑝 ) − 2(𝑞𝐵 − 𝑤)𝜆2 𝐹𝐶 = (𝑚1𝐶 + 𝑚𝐶 )𝑞𝐶̈ + 𝑔 (𝑚1𝐶 + 𝑚𝑝 ) − 2(𝑞𝐶 − 𝑤)𝜆3 𝑚𝑝. 𝑚1𝑘 𝑚𝑘. ∆𝑥𝐴 𝜆1 + ∆𝑥𝐵 𝜆2 + ∆𝑥𝐶 𝜆3 = (𝑚𝑝 + 𝑚𝐴 )𝑢̈ − 𝑓𝑢 ∆𝑦𝐴 𝜆1 + ∆𝑦𝐵 𝜆2 + ∆𝑦𝐶 𝜆3 = (𝑚𝑝 + 𝑚𝐵 )𝑣̈ − 𝑓𝑣 ∆𝑧𝐴 𝜆1 + ∆𝑧𝐵 𝜆2 + ∆𝑧𝐶 𝜆3 = (𝑚𝑝 + 𝑚𝐵 )𝑤̈ + 𝑔 (𝑚𝑝 + 𝑚𝐵 ) − 𝑓𝑣. ∆𝑥𝐴 = 2Δ𝑈S𝐴 ; ∆𝑥𝐵 = 2Δ𝑈S𝐵 ; ∆𝑥𝐶 = 2Δ𝑈S𝐶 ∆𝑦𝐴 = 2Δ𝑉S𝐴 ; ∆𝑦𝐵 = 2Δ𝑉S𝐵 ; ∆𝑦𝐶 = 2Δ𝑉S𝐶 ∆𝑧𝐴 = 2Δ𝑊S𝐶 ; ∆𝑧𝐵 = 2Δ𝑊S𝐶 ; ∆𝑧𝐶 = 2Δ𝑊S𝐶.
(213) 213.
(214)
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