工业机器人第三章欧拉角DH参数

1、.1.2.3.4( , )( , )( ,)( , )RPYRot zRot yRot x .50000100000010000( , )00100000000100010001cscssccsRPYsssc coscsins00( , )00001c cc s ss cc s cs cs s sc cs s cRPYsc sc c .600( , )00001xxxyyyzzznoanoaRPYnoa .722cosxynn22tan2(,)tan2(,)tan2(,)zxyxxyzAnnnAo nAaa.8.9( , )( , )( ,)( , )000010000001000000100

2、000000100010001EulerRot zRot yRot xcscssccsscsc coscsins00( , )00001c cc s ss cc s cs ss cs s sc cs s cc sEulersc sc c .10.11.12( , )( , )( ,)( , )00000000010000001000001000010001000100EulerRot zRot yRot zcscscsscscscc c cs sc c ss cc ss c cc ss c sc cs ss cs 00001sc .1300( , )00001xxxyyyzzznoanoaEu

3、lernoa 22tan2,tan2(,)tan2(,)xyzzzyxAaaaAo nAaa.14.15.16.17.18.19.20.21.22.23.24.25601 23456TTT TT TT.26.27600001xxxxyyyyzzzznoapnoapTnoap.28601 234560001xxxxyyyyzzzznoapnoapTTT TT TTnoap.29.30.31.32连连杆杆变变量量adcossin11 1-90000-122 20a2d21033 39000144 4-900d40-155 59000166 600d610.331122223311222 23312

4、3244556445545640000000000,0100001010000010001000100000000,010010000010001cscsa ccsscsca sscAAAdcscscscscAAAd666600000010000sscdcosiicsiniis.34cos()ijijc sin()ijijs5 65 656 55 65 656 5645666000001c cc ssd ss cs scd cTA Asc4 5 64 64 5 64 64 564 54 5 64 64 5 64 64 56 4 5634565 65 656 540001c c cs sc c

5、ss cc sd c ss c cc ss c sc ss sd s sTA A As cs scd cd.35111111116236311110001xxxxyyyyzzzznoapnoapTATnoap134 5 64 63 5 6134 5 64 63 5 614 5 64 6134 5 64 63 5 6134 5 64 63 5 614 5 64 613 4 53 513 4 53 514 513 4 563(),(),(),(),(xyzxyzxyzxnc c c cs ss s cns c c cs sc s cns c cc socc c cs cs s cosc c cs

6、cc s sos c cs cac c ss cas c sc cas spc c s ds 56413 4 56356414 56),(),yzc ddps c s dc c ddps s d.3622222222612345622220001xxxxyyyyzzzznoapnoapTA A A A Anoap2234 5 64 623 5 62234 5 64 623 5 624 5 64 62234 5 64 623 5 62234 5 64 623 5 624 5 64 6223 4 523 5223 4 523 524 52(),(),(),(),xyzxyzxyzxncc c cs

7、 ss s snsc c cs sc s sns c cc socc c cs cs s sosc c cs sc s sos c sc cac c ss cas c sc cas sp 23 4 562356422223 4 56235642 224 562(),(),yzc c s dsc dda cps c s dcc dda sps s dd.3761234560001xxxxyyyyzzzznoapnoapTA A A A A Anoap1234 5 64 623 5 614 5 64 61234 5 64 623 5 614 5 64 6234 5 64 623 5 61234 5

8、 64 623 5 614 5 64 61234 5 64 6()(),()(),(),()(),()xyzxync cc c cs ss s cs s c cc sns cc c cs ss s cc s c cc snsc c cs sc s coccc c ss cs s sss c sc coscc c ss cs 23 5 614 5 64 6234 5 64 623 5 6123 4 523 51 4 5123 4 523 51 4 523 4 523 51623 4 523 52342216 4 521623 4 523 5(),(),(),(),()(),()zxyzxys s

9、cs c sc cosc c ss cc s sac c c ss cc s sas c c ss cc s sas c sc cpc d c c ss cs da cs d s sdps d c c ss c 2342216 4 52623 523 4 52342 2(),()zs da cc d s sdpd c cs c cc da s.386010149.09001921.1210000001T.39.40连杆连杆 变量变量adcossin11 1-90000-122 2900d20133 300d31044 4-90000-155 59000166 600010.41cosiicsi

10、niis1122223311222 233123244556445545640000000000,0100001010000010001000100000000,010010000010001cscsa ccsscsca sscAAAdcscscscscAAAd666600000010000sscd.4261234560001xxxxyyyyzzzynoapnoapTA A A A A Anoap1234 5 64 623 5 614 5 64 61234 5 64 623 5 614 5 64 6234 5 64 623 5 61234 5 64 623 5 614 5 64 61234 5

11、 64 623 5()()()()()()()()xyzxync cc c cs ss s ss s c cc sns cc c cs ss s sc s c cc snsc c cs sc s soccc c ss cs s sss c sc coscc c ss cs s 614 5 64 6234 5 64 623 5 6123 4 523 51 4 5123 4 523 51 4 523 4 523 51 23121 231223()()()()zxyzxyzscs c sc cosc c ss cc s sac c c ss cs s sas c c ss cc s sas c sc

12、 cpc s ds dps s dc dpc d .43120001xxxxyyyynzzzznoapnoapA AAnoap.44.451234560000 xxxxyyyyzzzznoapnoapA A A A A Anoap.4611A11623456A TA A A A A1111610000001000010001xxxxyyyyzzzzcsnoapscnoapTnoap.4761T112xys pc pdcossinxypp22,tan2(,)xyyxppApp.4821221222212sin()cos()1()tan2,1ddddA 222122tan2(,)tan2(,)yx

13、xyAppAdppd.49112234 2322xyc ps pa cd sa c2 234232 2xpa sd ca s3 34 34 3a cd sd sk2222222232422xyzpppaaddka.5022233434tan2(,)tan2( ,)Aa dAkadk.511132A A1 231 23232 31 231 23232 363112000010001xxxxyyyyzzzzc cs csa cnoapc ss sca snoapTscdnoap1111610000001000010001xxxxyyyyzzzzcsnoapscnoapTnoap4 5 64 64

14、5 64 64 564 54 5 64 64 5 64 64 56 4 5634565 65 656 540001c c cs sc c ss cc sd c ss c cc ss c sc ss sd s sTA A As cs scd cd.521 231 2323233xyzc c ps c ps pa c a1 231 23232 34xyzc c ps c pc pa sd32 3112 342322112 341132 3232211()()()()()()()()zxyzxyzxyzxyaa cpc ps pa sdspc ps pa cdpc ps paa sspc ps p.

15、53232332 31142 32 341132 3tan2()()()()()()xxyzxyAaa cpc ps pda sa sdpc ps paa c2233.541 231 234 5114 5xyxya c ca s cc sa sa cs s 4111 231 2323tan2(,)xyxyzAa sa ca c ca s ca s.55111432A A A1111610000001000010001xxxxyyyyzzzzcsnoapscnoapTnoap11114321664A A A A TT1 23 41 41 23 41 423 42 3 42 43 41 23 41

16、 41 23 41 423 42 3 4243 4641 231 23232 340001c c cs ss c cc ss ca c cd sa cc c ss cs c cc ss sa c sd ca sTc ss sca sod.565 65 656 55 65 656 5645666000001c cc ssd ss cs scd cTA Asc1 2341 41 2341 422345()()()xya c c cs sas c cc sas cs 1 231 23235()()()xyzac cas sacc555tan2( ,)As c.571111154321665A A A

17、 A A TT61 23 41 41 23 41 423 4()()()xyzsn c c ss cns c sc cn s s 61 23 41 451 23 51 23 41 451 23 423 4 523 5()()()xyzcncc cs s cc s sns c cc s cs s cn s c cc s666tan2(,)As c.58.59.6011121314212223243132333441424344aaaaaaaaTaaaaaaaa13111214232122243132333443414244aaaaxxxxaaaaxxxxdTdxaaaaxxxxaaaaxxxx.

18、61(,)( ,)TdTTrans dx dy dz Rot k dT(,)Trans dx dy dz( ,)Rot k d(,)( ,)dTTrans dx dy dz Rot k dI T(,)( ,)Trans ds dy dz Rot k dI .62(,)( ,)Trans ds dy dz Rot k dI 100010(,)0010001dxdyTrans dx dy dzdz1010( ,)100001zyzxyxk dk dk dk dRot k dk dk d.631001010000101001000011000100001000100010000000zyzxyxzy

19、zxyxdxk dk ddyk dk ddzk dk dk dk ddxk dk ddyk dk ddz 0000000zydxzxdyyxdz 1.6400110100501000001A00.5dijk 00.10ijk.65000.1100000.1000.50000 0000000zydxzxdyyxdz 00.5dijk 00.10ijk00.10 xyz100.5dxdydz.66000.1100110000010050.1000.501000000000100.1010000000.10.50000dAA .67TdTTT 1TTT dTT TdTTT.68 这个变换方程如同前面

20、变这个变换方程如同前面变换方程一样，可以用一个变换换方程一样，可以用一个变换图来表示，如右图所示。由图图来表示，如右图所示。由图也可以直接得到上式。也可以直接得到上式。 方程方程2 2很重要，因为它把相很重要，因为它把相对于不同的坐标系之间的微分对于不同的坐标系之间的微分变化联系起来了。我们首先展变化联系起来了。我们首先展开方程右端的矩阵乘积，展开开方程右端的矩阵乘积，展开过程中进行了简化，可得出微过程中进行了简化，可得出微分变化向量分变化向量d d和和的元素之间的的元素之间的直接关系。变换直接关系。变换T T称为微分坐标称为微分坐标变换。变换。1TTT 2.69如果把微分坐标变化如果把微分坐

21、标变化T T的元素用向量的元素用向量n n、o o、a a和和P P描述为描述为0001xxxxyyyyzzzznoapnoapTnoap()()()()()()()()()()()()0000 xxxxyyyyzzzznoapdnoapdTnoapd.701()()()()()()()()()()()()0000nnnonanpdonoooaopdTTanaoaaapd式中，式中，d d和和就是微分旋转和微分平移。将上式左乘就是微分旋转和微分平移。将上式左乘 可得可得1T.710()()()()0()()()()0()0000Tn oanpnd nn oo ap od oano apad a

22、 ()()()ab cbacbca ()0aac.72n oaanoo an0()0()0()0000Taopnd nanp od oonpad a 0000000TTTTTTTTTTzydxzxdyyxdx 由于由于n、o、a正交，所以正交，所以34由式由式1定义为：定义为：T.73 令式令式3 3和式和式4 4相等，我们就利用相对基础坐标系来描述的相等，我们就利用相对基础坐标系来描述的微分旋转和平移的向量微分旋转和平移的向量( (和和d)d)，得到相对于坐标系，得到相对于坐标系T T来描述的来描述的微分旋转和平移的向量，即微分旋转和平移的向量，即()()()TTTdxpnd ndyp od odzpad aTTTxnyoza65.74() ()()()() ()()()().TxyzxyzxyzxyzxxyyzzxxyyzzdxijkPn iPnjPn kd id jd kn in jn kPnPnPnn dn dn d()()(),()()(),.Txxyyzzxx