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机械臂运动规划基础知识.pdf

发布时间：2022-05-30 发布人：admin 分类：说明书 资料大小：0.48M 资料格式：pdf 举报 版权申诉
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    ¯<˘£o( ø 2019c7 1 ~^^= R(X, θ) = R(Y, θ) = 0 0 cos(θ) − sin(θ) cos(θ) 0 sin(θ) 0  1  cos(θ)  cos(θ) − sin(θ) cos(θ) sin(θ) 0 0 1 − sin(θ) 0 cos(θ) sin(θ) 0 0 0    R(Z, θ) =  kxkxversθ+cθ 0 kxkyversθ − kzsθ kxkzversθ + kysθ kykzversθ − kxsθ kykxversθ + kzsθ kykyversθ+cθ kzkxversθ − kysθ kzkyversθ + kxsθ kzkzversθ+cθ 0 1 R(K, θ) = “¥versθ = 1 − cos(θ), sθ = sin(θ), cθ = cos(θ), K = (kx, ky, kz)T 2 gIgC BIXØuAIXgCXe A BT = gIk–eEC’X A BR APBO 0 1 A CT =A B T B C T 1  (1) (2) (3) (4) (5) (6)
    liang-j14@tsinghua.org.cn 3 IC gCECƒ^SC§ƒ{˘"/ˇgC:ICC AP =A B T P B Av =A B T vB (7) (8) :ICC«O·14'0§:I14'1. 'fi†7?L:¶K^=θ^=Cˇ“R(K, θ)§y3–/ˇøgCˇ“ ƒ7?L:P (px, py, pz)¶K^=θgCˇ“§øgCˇ“ R(K, θ) −R(K, θ)P + P A BT = T rans(P )R(K, θ)T rans(−P ) = 0 1 (9) 3 IC dugIC–?1E§ˇd§ØuXGØIX5‘§⁄kIX C’X§U?¿IXmC’X"øC’X~k^§3¯:Œ˘L§ ¥§|^IC–ØN·/ƒ¯:"3˜IX¥’u’!C…ŒL“§/ˇø …ŒL“–?1"$˜"y3b‰kIX§˜IX§Ψ0$˜IXΨd"ø IX3—'·›" 3.1 ؉IX$˜ 3ø«„e§zgC·Øu˜IX§’X7Y¶^=θ§øp¡Y¶ ·˜IXY¶"øAT#gC“3EC“?1ƒ§·‘§ l˜IXCICAT3m>§3>"·CE,·0 nT "ø¿ X§"IX¥‰IAT3Cm?1ƒ§ø·§3˜IX¥ L«"= 0P = (n−1 n T . . .1 2 T ·0 1 T ) ·n P =0 n T ·n P (10) 3.2 Ø$˜IX$˜ «„·§ø«„ezgC·Øu#IX(·$˜IX)§ ’X7Y¶^=θ§øp¡Y¶·$˜IXY¶§2·˜IXY¶"øAT #gC“3EC“m?1ƒ§·‘§l˜IXCICAT 3>§3m>"dC·0 nT "ø¿X§"IX¥‰IAT 3Cm?1ƒ§ø·§3˜IX¥L«"= 0P = (0 1T ·1 2 T . . .n−1 n T ) ·n P =0 n T ·n P (11) 2
    liang-j14@tsinghua.org.cn 3 IC 3.3 “IX$˜ k¡«˜„§–#IC·Øu˜IX§ATr#IC “3¶XJ#IC·Øu$˜IX(gCIX)§ATr#I C“3m"3XIC“ºƒ¥–3Øu‰IX$˜C“Øu $˜IX$˜C“§5¿§{^S="y3/ˇmatlab5?1y"b‰$˜IX ¥k:(1,1,1)§k4ƒ7Ψ0Z¶=− π §24§7Ψ0Y¶=arcsin( 1√ )"d– 0.8165 0 0.5774 0 0 1 0 0 −0.5774 0 0.8165 0 1 0 0 0 0.7071 0.7071 0 0 −0.7071 0.7071 0 0 1 0 0 0 0 0 0 1  0P =  0P = 4      3  1 1 1 1  =  1.7321 0 0 1    1 1 1 1  =  1.7321 0 0 1  (12) (13) (14) w,ø(J·("y3–$˜IXº›#LªøCL§"KdL§lI› )§¥K¶·Ψ0Y¶3Ψd¥L“"u· ¶27ΨdK¶=arcsin( 1√ 3 m'§k7ΨdZ¶=− π – 4 0.7071 0.7071 0 0 −0.7071 0.7071 0 0 1 0 0 0 0 0 0 1 0.9082 −0.0917 0.4082 0 −0.0917 0.4082 0 −0.4082 −0.4082 0.8165 0 1 0.9082 0 0 0 XJƒ§‹uy«„e·"ˇLmatlabyUuy§3 ª«„egC“2E#IC“§ˆø#IC“·–«“¥= «ogC“E§#ogC“·"?–§•’u“ IX$˜·(" 3.4 _IC fiICA BT §XJƒAØuBL«KIƒA BT _= A BRT ·A PBO BRT −A 0 1 AT =A B B T −1 = 3.5 .RPY 3.5.1 RPY RPY·£ªE˝3¥˚1^«{"E1¤Z¶§K7Z¶^=(α)¡ E˜(Roll)¶r7Y¶^=(β)¡:(Pitch)¶rYX¶§7X¶^=(γ) ¡ =(Yaw)"ø«L«{‰k7˜IXX¶^=§27˜IXY¶^=§27˜IXZ¶^ =§¢S·«7‰IX$˜“"u·k A BRXY Z(γ, β, α) = R(ZA, α)R(YA, β)R(XA, γ) (15) 3
    liang-j14@tsinghua.org.cn  cαcβ cαsβsγ − sαcγ cαsβcγ + sαsγ sαcβ sαsβsγ + cαcγ sαsβcγ − cαsγ −sβ cβsγ cβcγ cos(β) = r2 11 + r2 21  =  r11 r21 r31  r12 r22 r32 r13 r23 r33 ƒ)ª“§–RPY^==’X XJcos(β) = 0§@ok β = arctan( −r31r2 11 + r2 21 ), α = arctan( r21 r11 ), γ = arctan( r32 r33 ) XJβ = π 2 §@ok XJβ = − π 2 §@ok 3.5.2 Z-Y-X. β = π 2 , α = 0, γ = arctan( r12 r22 ) β = − π 2 , α = 0, γ = − arctan( r12 r22 ) 3 IC (16) (17) (18) (19) (20) ZYX.˜k7$˜IXZ¶=α§,7$˜IXY¶=β§7$˜IXX¶ =γ"ø¢S·«7$˜IX$˜IC§ A BRXY Z(γ, β, α) = R(ZB, α)R(YB, β)R(XB, γ) (21) ø(J«{·§ƒ{§·Ø(J) ”" 3.5.3 Z-Y-Z. ø«{7$˜IX?1$˜"k7$˜IXZ¶=α§27$˜IXY¶=β§q7 $˜IXZ¶=γ"u· A BRZY Z(α, β, γ) = R(ZB, α)R(YB, β)R(ZB, γ)  cαcβcγ − sαsγ −cαcβsγ − sαcγ cαsβ sαcβcγ + cαsγ −sαcβsγ + cαcγ sαsβ cβ −sβcγ sβsγ  =  r11 r21 r31  r12 r22 r32 r13 r23 r33 (22) (23) (24) (25) (26) r2 XJsin(β) = 0§@ok β = arctan( XJβ = 0§@ok XJβ = π§@ok 31 + r2 32 r33 ), α = arctan( r23 r13 ), γ = arctan(− r32 r31 ) β = 0, α = 0, γ = arctan(− r12 r11 ) β = π, α = 0, γ = arctan(− r12 r11 ) 4
    liang-j14@tsinghua.org.cn 4 ¯<$˜5y 4 ¯<$˜5y 4.1 DHºŒ º\dºŒ5L«·º\ai§L«’!i^=¶’!i+1^=¶m l¶,·º\αi§L«’!i^=¶=’!i+1^=¶)Y"^’!¶º X^œ{§'OØAº\iº\i+1"ø^œ{mldi§L«º\¶ø^œ{ mYθi§L«’!"øºŒ·’!ºŒ"3D˜§S.5‰ a0 = an = 0, α0 = αn = 0 (27) XJ’!i·^=’!§@oθi·’!C§di‰C§‰di = 0"XJ’!i·£˜’!§@odi ·’!C§θi‰C§‰θi = 0" d⁄ª§zº\d4ºŒ⁄£ª§¥£ªº\§,£ªº\º\º ’X"Øu^=’!§θi’!C§ƒ3‰C§¡º\ºŒ¶Øu£˜’!§di’!C§ ƒ3‰C§¡º\ºŒ"ø«£ª¯$˜’X{¡DH{§’ºŒDHºŒ" 4.2 º\C ˜kƺ\IX§/ˇº\IXDHºŒº\CL“"º\IXX¶ œ{§i-1i¶º\IXZ¶’!¶§g(DH{¥ƒb)¶º \IXY ¶dmˆIX(‰"3ø«„e i−1 i T = Rot(X, αi−1)T rans(X, ai−1)Rot(Z, θi)T rans(Z, di)  cθi sθicαi−1 sθisαi−1 0 −sθi cθicαi−1 −sαi−1 −disαi−1 dicαi−1 cθisαi−1 cαi−1 ai−1 0 0 0 1  i−1 i T = 4.3 ¯<$˜˘§ (28) (29) (30) d’!C‰´§XJ^–$˜IXº“5ƒIXmgIC§ K–ØN·/¯:"’u’!C$˜˘§"= 0 nT =0 1 T 1 2 T . . .n−1 n T = 0 nR 0Pno 0 1 ø$˜˘§ºm’!m§ƒ3m–=z’!m§Or <¯pl—5" 5
    liang-j14@tsinghua.org.cn 5 '$˜’ 5 '$˜’ fiA!BIXIC§–9,'$˜3A¥L“§ƒT'$˜3B¥L“" '$˜C’X3¯<'!•'!˜˘'¥k›^"P'£˜'=˜ ¥'O - S(P ) = KØu?¿IXA!B§'$˜¥dC d = [dx, dy, dz]T , δ = [δx, δy, δz]T  0 pz −py −pz 0 px py −px 0 Bd Bd Bδ = A B = Bδ BRT −A 0 AR −B 0 BRT S(APBO) BRT A ARS(APBO) B AR Aδ  Ad Ad Aδ |^’–m"$˜C’!m"C’XXe [υ, ω]T = J(q) ˙q = J(q)[ ˙q1, ˙q2, . . . . ˙qn]T  J(q) =  ∂Px ∂θ1 ∂Py ∂θ1 ∂Pz ∂θ1 ∂ωx ∂θ1 ∂ωy ∂θ1 ∂ωz ∂θ1 ∂Px ∂θ2 ∂Py ∂θ2 ∂Pz ∂θ2 ∂ωx ∂θ2 ∂ωy ∂θ2 ∂ωz ∂θ2 . . . . . . . . . . . . . . . . . . ∂Px ∂θn ∂Py ∂θn ∂Pz ∂θn ∂ωx ∂θn ∂ωy ∂θn ∂ωz ∂θn (31) (32) (33) (34) (35) (36) ’k,«ƒ“"-n, o, α, P 'Oi nT 1,2,3,4o"XJ’!i^=’!§ K JLi = [−nxPy + nyPx,−oxPy + oyPx,−αxPy + αyPx, nz, oz, αz]T XJ’!i£˜’!§K JLi = [nz, oz, αz0, 0, 0]T (37) (38) 6
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