机械臂关节空间B样条曲线轨迹规划_王幼民.pdf

1、15 2 Vol.15No.22000 M6Journal of Anhui Institute of Mechanical Electrical Engineering Jun.2000cI|:1007-5240(2000)02-0021-06 D1 bWB“H wLE?王幼民(“, 241000)K 1:1 bW PB“H wLDE?,is 3Q“H wL1 ,B“HZEE? s ) 、9 y、#9 =i .1oM:D;B“Hf ;E?ms |:TP241.2 DS M :A- DE? Vs 5: bW1 bW? , - Q4,L ,ZE #v 5: bW1 bWv, p I ,9 v,i

2、 O . ?D 5:US“ , 4 L H e. 5:US bW/E? , P _1 bWE?ZE$.D1 % K HW5,ZE B.D24B HWK E?ZE,ZE1 bWi I n K 、L= ,ZE , O ? L.D3、44 CPE Q“Hf ZE,ZE V1? p,W n H,NE9 v, Oi I n .D44 B CPE? E, E 1;i =0,1,n -k;u .(1) T, VwB“Hf .k =4 H, V p QB“Hf :Bi,4(u)=13!1 u u2 u31 4 1 0-3 0 3 03 -6 3 0-1 3 -3 1(2)1.2 Q ( B“Hf wZE!f

3、Vr T(i ):i(u)=X0(u)Vi-1+X1(u)Vi +X2(u)Vi+1+X3(u)Vi+2 (3)T(3),u ,0 u 1;X1(u) T,i(u)V Ui B“H wL i O ;Vi-1,Vi,Vi+1,Vi+2B“H wLBF e. 1 p,M # B“H wLi(u)i+1(u)sYu =0u =1) / Hq:i(1)=i+1(0). (4)| T(4) T(3), V:X0(1)=X3(0)=0,X1(1)=X0(0),X2(1)=X1(0),X3(1)=X2(0). (5)“,M # wL ) | q w q, / Hq,:i(1)=i+1(0),1(1)=i+1

4、(0).N V/ 8Hq, :X0(1)=X3(0)=0,X1(1)=X0(0),X2(1)=X1(0),X3(1)=X2(0),X0(1)=X3(0)=0,X1(1)=X0(0),X2(1)=X1(0)=0,X3(1)=X2(0) (6)N, O1“,USMMHq:X0(u)+X1(u)+X2(u)+X3(u)=1 (7)L !X(u) Q T, Hq V p Q T “ . I nf , VL !:X0(u)=X3(1-u), X1(u)=X2(1-u),X2(u)=a20u3+a21u2+a22u +a23,X3(u)=a30u3+a31u2+a32u +a33.X0(1)=X3(0)=

5、0,X0(1)=X3(0)=0,X0(1)=X3(0)=0.yN V:a23 =a30,a22 =3a30,a21 =3a30.“ V:a31 =a32=a33 =01“, (7) T, I n,a20=-3a30,a30 =1/6, V pXi(u) /:232 X0(u)=(1-u)3/6,X1(u)=u3/2+2/3-u2,X2(u)=-u3/2+u2/2+u/2+1/6,X3(u)=u3/6.1.3 QB“H wL QB“H wL C2) , QB“H wL OZ:i(u)=3j=0Bj,3(u)Vi+j-1 (8)T(8):i =1,2,m -1;m1 bW ,V e.T(8) T:

6、i(u)=(1/6)1 u u2 u31 4 1 0-3 0 3 03 -6 3 0-1 3 -3 1Vi-1ViVi+1Vi+2(9)T(9)i =1,2,m -1. QB“H wL / :B: .i B“H wLi(u)(0u 1) S O sY:i(0)=(1/6)(Vi-1+4Vi +Vi+1),i(1)=(1/6)(Vi +4Vi+1+Vi+2).B ( )= ,yB“H wLB O= OsY:i(0)=(Vi+1-Vi-1)/2 i(1)=(1/2)(Vi+1-Vi-1),i(0)=Vi+1-2Vi +Vi+1 i(1)=Vi +2Vi+1 -Vi+2. (10) T(10) V

7、B、= O .i+1(0)=Vi -2Vi+1+Vi+2 =i(1);i+1(0)=(Vi+2-Vi)/2.=: . QB“H wL M # e ,yNM+H ,Y1M # wL, Z wL M.BDE HVC,D E V,“ 8 HW, D L H e.1.4 e pE(9) TVi-1、Vi、Vi+1、Vi+2 BF e,DE?,5 pE ,D I5 p p .1E?,A5 p e. !P1 bW,/ Hq:i-1(1)=i(0)=Pi (i =1,2,m -1) (11)7 i-1(1)=i(0)=(1/6)(Vi-1+4Vi +Vi+1) (12)T(12)m+2 ,mZ,yN1 Hq

8、, HHq.HHq) n, |Hq,V1 =V0,Vm+1 =Vm.6 -61 4 11 4 11 4 16 -6V0V1V2VmVm+1=60P1P2Pm0(13) D1 bWB“H wLE?242000 MZ(13) VB BFV0、V1、V2、Vm、Vm+1.Vi-1、Vi、Vi+1+Vi+2 B“H wL:i(u)=(1/6)u3 u2 u 1-1 3 -3 13 -6 3 0-3 0 3 01 4 1 0Vi-1ViVi+1Vi+2(14)i(u)=(1/6)(-u3+3u2-3u +1)Vi-1+(1/6)(-u3+6u2+4)Vi +(1/6)(-3u3+3u3-3u +1)Vi

9、+1 +16 u3Vi+2 (u 0 u 1) (15)!1 bWB“ P1、P2、Pm,Z(13) V pm +2 eV0、V1、V2、Vm、Vm+1, M # W VBHB“H wL| ,Em-1 B“H wL 7, !i B“H wL Pi、Pi+1 ,iwLVi-1、Vi、Vi+1、Vi+2 e e, !4 eUS:Vi-1(ti-1,qi-1)、Vi(ti,qi)、Vi+1(ti+1,qi+1)、Vi+2(ti+2,qi+2).i B“H wL T(14). T,u ,i(u) wL , uR |01W, ViwL.iwL i、:USsYt(u)、q(u)V U.t(u)=(1/6)

10、u3 u2 u 1-1 3 -3 13 -6 3 0-3 0 3 01 4 1 0ti-1titi+1ti+2(16)q(u)=(1/6)u3 u2 u 1-1 3 -3 13 -6 3 0-3 0 3 01 4 1 0qi-1qiqi+1qi+2(17)1.5 HHq pV0、Vm+1,B“HE?1F2HHq,B“HZEHHqVC F eV0#Vm+1.(14) T H H u p,EB O= O, sY|(u),(u)V U:i(u)= ddui(u)=(1/6)(-3u2+6u -3)Vi-1+(1/6)(9u2-12u)Vi +(1/6)(-9u2 +6u +3)Vi+1+u2V1+2

11、/2 (18)i(u)= ddui(u)=(1/6)(-6u2+6)Vi-1+(1/6)(18u -12)Vi +(1/6)(-18u +6)Vi+1 +uVi+2 (19)i(u)=di(u)du = dt(u)du , dq(u)du = t,q (20)(20) Tt dt(u)du ,q dq(u)du )252 i(u)=di(u)du =dtdu ,dqdu = t,q (21)(21) T:t=dtdu ,q=dqdu.Vr T:i(u)=q/t (22)F Vr T:i(u)=d(u)dt =di(u)du dudt =qt-qtt3 (23) 、F Vr T V,1B O ,

12、 H,1 ,;1= O , H,1F ,.1B O , B OBs , s ,B O l(1,0).1) 、 ,.1(0)=(1/6)(-3)V0+(1/2)V2 = V0 =V2 -21(1)=(1/6)(-3+6-3)Vm-2 +(1/6)(9-12)Vm-1+(1/2)Vm+1 = Vm+1 =2+Vm-1.2) ,F ,.1(0)=V0-2V1+V2 =0, V0 =2V1-V2.m+1(1)=Vm-1-2Vm +Vm+1 =0, Vm+1 =2Vm -Vm-1.3) ,F ,. ZE V pHHq:V0 =V2-2, Vm+1 =2Vm -Vm-1.4) F , ,.V0 =2V1-

13、V2, Vm+1 =2+Vm-1.5).tHHq,Hq:V0 =V1, Vm+1 =Vm. HHq V“ZEw.m1 1B“HE?m2 E? D ( m) 5: bWVB 6B(L wL), S#EWYV I5 p1 bW,1 bW VYVD6ZE p.Puma560 ,D3 S Puma560 - B“HE?,D3 S V1,WHWWnV2, ti -ti-1=hi-2(i =39).E wLnm1.V1,110F, SHq、HHq p,HHqV2 =Vo =0. D1 bWB“H wLE?262000 MV1 V1 2 31 2 31 81 19 2202 10 15 453 60 25

14、1804 75 30 2005 130 -45 1206 110 -55 157 100 -70 -108 -10 -10 1009 -50 10 5010 133 103 -22V2 D3 HWWVs HW HW HW HW HW HW HWh1 3.472 h2 2.274 h3 2.714 h4 3.664 h5 1.761 h6 3.971 h7 3.9083 T) YV s, H9 _ V / :1)B“HZEE wL HWB、= ,1 bW 、F .2)B“HE1MM ql.rE m 3、 .3) .B“HZEE g, gDE 31Y,C ? T M,t Y -,1pDE? ?s

15、、 y 、 ? TE,7B“HZEDE? ?、 E4 ZL. ID:1 Roth.B, and Kakn.M.EThe Near-minimum-Time control of Open-Loop articulated Kinematic ChainsJ .Trans.AME.J.Dynamicsystems, Measurement and Control.1971,93:164-172.2 Kim BK, Shin KG., An Efficient Minimum-Time Robot Path Planning Under Realistic comditionsC .Procof1

16、984 American Control conference, san dieg, 1984:296-303.3 Lin ES, Chang PR, and Luh J.Y.S.formulation and Optimization of Cubic Polynomial Joint Trajectories for In-dustrial RobotsJ , IEEE Trans.On Automatic Control.1983, 28(13):1066-1073.4 _ , P. E? EJ. ,1988,2(6):18-25.5 V. / #M.:t bt?v,1994.142-1

17、46.6 . 5# I5M. 2 v.1991.B-splines path planning for manipulator in jiont coordinatesWANG You-ming(Mechanical Engineering Department of Anhui Institute of Mechanical and Electrical Engineering, Anhui Wuhu 241000, China)Abstract:B-Splines functions are applied to path planning for robots in jiont coordinates and com-paredwith themethoelof segmentalcubicsplinefunctions.Theadvanteges of themethod are that it canbe handledsegmentally, speed up computation and take up less internal memory.Key words:manipulator, B-splines, pathplanning