1、1R4R3R2R2 4 3 0iR 1R4R3R2R2 4 3 2314RRRR2233114422331144r cr crcr cr sr srsr s 2 223 334 442 223 334 44r sr sr sr cr cr c 10 3 334 442 223 334 442 22r sr sr sr cr cr c 3 334 442 223 334 442 22r sr sr sr cr cr c 334432 22334442 22r sr sr sr cr cr cCD 22233442 222 223 334 443222334442 222 223 334 44r
2、sr sr sr cr cr cr cr cr cr sr sr s AB 1A B 运运动动学学方方程程dtdtdtdtdtdt运运动动输输入入加加速速度度速速度度位位移移Txyzppp oxyzPTuvwppp ouvwPxuvwyuvwzuvwp(ppp)p(ppp)p(ppp)ouvwxuvwxouvwyuvwyouvwzuvwzPiiikiPjiikjPkiikkxyzpppuxvxwxuuyvywyuuuzvoxyzouzwvwziijikipijjjkjppikPRjkPkkuxvxwxuyvywyuzvzwziijikiRijjjkjikjkkkuvwpppouvwuvwPij
3、k1ouvwoxyzTQRPQPR,oxyzxouvwPRP,1000cossin0sincosxxuRii,zyxRRRR ,TuvwRRRR ,cos0sincos-sin0010sincos0-sin0cos001yzRR cos0sincos-sin0100010sincos00cossin-sin0cos0010sincos0010-1231221331010001010000-1-100001010oxyzP 1000cossin0sincos110011200-123301032oxyzP cos-sin0100sincos00cossin0010sin10-1010013210
4、000-12130010cos1032oxyzP 0AABABBpRpp0BABABAppRp，复合变换方程为：因为在一般变换过程中非齐次的。1100010ppRpBBAABA的列向量14X4X4的方阵 TAB设为：pTpBABA可简化为：为齐次变换矩阵。称齐次变换。TAB110BABABAppRp的特点：TAB10000BAABABpRT坐标坐标BB相对于相对于AA的的旋转矩阵（旋转矩阵（3X33X3）坐标坐标BB的原点在的原点在AA坐标系中的坐坐标系中的坐标。（标。（3X13X1）的位置和方位。相对于描述了坐标系：即ABTAB标示符，标示符，“1”1”位置；位置；“0”0”方向方向的另一种
5、变形：齐次变换矩阵 TAB10000BAABABpRT3 30000010001AAxBBIpR ,xy,z1000c0s00c-s001000sc0-s0c000010001100dc-s00010dsc000010001d00010001xyztranTTTT 03 300000100010001AAAAABBxBBBRpIpRT ，有：，则对于空间任一点的描述为相对于，的描述为相对于；、坐标系已知三维空间中的三个pTBCTABCBABCABpTpCBCBpTpBABApTTCBCABpTCACTTTBCABAC的描述）相对于复合变换（AC。，到达最终作运动相对于，然后，到达作运动重合，首
6、先相对于与是这样得到的：最初坐标系：这种变换的另一种解释CTBBTAACCBCAB系而言的。：运动相对于运动坐标“从左到右”右乘系而言的。：运动相对于固定坐标“从右到左”左乘变换顺序)()(AB11 111BC22222(l cq,l sq,0)(z,q)(l cq,l sq,0)(z,q)TTransRotTTransRot1q2q0 x0y1l2l1y1x2y3x111111110cosq-sinq0l cosqsinqcosq0l sinq001000010001AAABBBRpT222222220cossin0cossincos0sin001000010001BBBCCCqqlqqql
7、qRpT1q2q0 x0y1l2l1y1x2y3xAABCBC1111222211112222111212122122121112121212212212cq-sq0l cqcq-sq0l cqsqcq0l sqsqcq0l sq0010001000010001cq cq-sq sq-cq sq-sq cq0l cq cq-l sq sq+l cqsq cq+cq sq-sq sq+cq cq0l sq cq+l cq sq+TTT21l sq00100001p11212p11212x=l cq+l c(q+q)y=l sq+l s(q+q)p11212p11212x=l cq+l c(q+q
8、)y=l sq+l s(q+q)p1121212122p1121212122x=-l sq-l s(q+q)q-l s(q+q)qy=l cq+l c(q+q)q+l c(q+q)q2iiiii22LLLLL2d (t)d(t)T(t)=J+B+dtdtd (t)d(t)iT(t)=J+B+T(t)dtdtT(t)2iiiii2LLLLLT(s)=J s (s)+Bs(s)+i=J s (s)+B s(s)T(s)T(s+T)(s)LiLiLiL2211=T-Ti11sJ+Js+B+BisiLiLee11=T-TJ s+Bii 20eLLJ=m22iieie2d (t)d(t)J=T(t)-B
9、dtdt2eiieiiieeJ s (s)=T(s)-B s(s)T(s)(s)=s J s+B0ieeLT(s)L(s)=2 s J s+B2iiiL2iL2222LLiiiLiL2i1iT(s)-1+J s+BsT(s)Kr(s)=1J s+BJ s+B s+i J+Js+i B+BsKr2iiiL2iL22LLiiiLiL2i1iT(s)-1+J s+BsT(s)Kr(s)=1J s+BJ s+B s+i J+Js+i B+BKr2iL2iLLiL1T(s)-T(s)ii J+Ji B+B(s)=s+iiii=1,2,L,ntqqidLLFd22211 11 111111111122co
10、s Km vm dPm gym gd222 2221,2Km vPmgy222222211212211212211 121212211 121212sinsin()coscos()coscos()()sinsin()()vxyxddyddxddydd222 2222222211211 221 2211 22211221212122cos2coscos Km vmddd dPm gdm gd1212KKKPPPLKP2222212112211 2222 1 2211 21211221211()(2)22cos()()coscos()Lmmdm dm d dmmgdm gd121122121221
11、21122122 1 22121()sinsin()()()cos(2)Lmmgdm gdLmm dm dm d d22121222 1 221122222 1 2222 1 22 1 22 1 22()2cos(cos)2sinsin dLmmdm dm d ddtm dm d dm d dm d d222122 1 22 1222222 1 2212222 1 221 2222122()cos(cos)sinsin()Lm dm d ddLm dm d dm dm d ddtLm gd1112221212 22 1 2212 22 1 22222 1 221 22 1 2221211221
12、2222222 22 1 2212 2 22 1 22coscos2sinsinsinsincossin d LLTdtmm dmdmddmdmddmddmddmm gdmgddLLTdtmdmddmdmdd2212212sinmgd()(,)()FD q qH q qG qT()(,)()eD q qH q qBqG qJ F niiiii 1m0Far12kqq.qti=1,2,nrriikijj 1jqqrris=jkk(s)iisiss 1s 1sqqqrrvkiiiijj 1jq,i=1,2,nqtrrvr n()iiiiFmv10siak(s)iss 1qirv*01,2,ssFF
13、sk()()1111(m)0(m)0 FavFavnk mk mnssiiiisiiiisissiiijCijivv ()()()()()()111uuuiiikkkstststijijsijCsiisiCCsssvvvvvv()()()()()()1u0iikssstttijCijisijCijisvvvv ()()()1u0iksssijCijissvv ()()()1,2,isssijCijivvsk11,2,nsisiFFsk()sisijijjFFv()()issisCiijijijjjFvFF iijiijijjjFFLF()()issisiCiiFF vL()()1insssiC
14、iiiFF vL FV FW L12121212121212(1)(1)(1)(2)(2)(2)12()()()(1)(1)(1)(2)(2)(2)12()()()nnnnnnTkCCCTCCCnkkkCCCCCCTCCCnkkkCCCFFFFvvvvvvVFFFFvvvWLLLL*11,2,nsisiFFsk*()msisijijijjFav*()()CmmissisijijijijijijFava *mMmiiijijiCjiijijijjFaaLa ()()()isssijCijivv *miijijijiiiiijLaII *()*()issisiCiiFFvL*()*()1insss
15、iCiiiFFvL*FV FW L12*12*12111111222222*MMM()()()nTkTCCnCnnnnnnFFFaaaeJJeJJeJJ FFL*0ssFF*FV FW LFV FW L*()()0VFFWLL2.2.广义变量广义变量Pe f()()()()()()tttEe t q tEe t f t dtEf t p t()()ttEf p dpEe q dq()()()()11()()()()q ti tx tv tu tq tFkx tx tCC0()()(0)1()()q tf t dtqe tq tC()()(0)1()()p te t dtef tp tI()()
16、()()11()()()()p tF ttu tv tp ti ttmL/eRffe R2112fnfeneTFn 1e1f2e2fGYr 1e1f2e2f2112erfTriurerf 12310nniiieeeea f12310nniiiffffa e元件1元件2元件1元件2ef元件1元件2元件1元件2feSeefSfefCefCef0001()(0)()()()te tef t dtCde tf tCdt0001()(0)()()()tf tfe t dtIdf te tIdtRefRef00()()1()()e tR f tf te tRTFn1e1f2e2fTFn1e1f2e2f12211/1/en efn f2112enefnfGYr1e1f2e2fGYr1e1f2e2f2112erferf2112/ferfer由全积分因果关系键图模型列写状态方程由全积分因果关系键图模型列写状态方程
