Calculating Jacobian coefficients of primitive constraints with respect to Euler parameters

Transcription

1 Calculatng Jacoban coeffcent of prmtve contrant wth repect to Euler parameter Yong Lu, Ha-Chuan Song, Jun-Ha Yong To cte th veron: Yong Lu, Ha-Chuan Song, Jun-Ha Yong. Calculatng Jacoban coeffcent of prmtve contrant wth repect to Euler parameter. Internatonal Journal of Advanced Manufacturng Technology, Sprnger Verlag, <hal > HAL Id: hal Submtted on 19 Dec 2013 HAL a mult-dcplnary open acce archve for the depot and demnaton of centfc reearch document, whether they are publhed or not. The document may come from teachng and reearch nttuton n France or abroad, or from publc or prvate reearch center. L archve ouverte plurdcplnare HAL, et detnée au dépôt et à la dffuon de document centfque de nveau recherche, publé ou non, émanant de établement d enegnement et de recherche frança ou étranger, de laboratore publc ou prvé.

2 Int J Adv Manuf Technol : DOI / ORIGINAL ARTICLE Calculatng Jacoban coeffcent of prmtve contrant wth repect to Euler parameter Yong Lu Ha-Chuan Song Jun-Ha Yong Receved: 5 March 2012 / Accepted: 12 November 2012 / Publhed onlne: 22 December 2012 Sprnger-Verlag London 2012 Abtract It a fundamental problem to calculate Jacoban coeffcent of contrant equaton n aembly contrant olvng becaue mot approache to olvng an aembly contrant ytem wll fnally reort to a numercal teratve method that requre the frt-order dervatve of the contrant equaton. The mot-ued method of dervng the Jacoban coeffcent to ue vrtual rotaton whch orgnally preented to derve the equaton of moton of contraned mechancal ytem. However, when Euler parameter are adopted a the tate varable to repreent the tranformaton matrx, ung the vrtual rotaton wll yeld erroneou formulae of Jacoban coeffcent. The reaon that Euler parameter are ncompatble wth vrtual rotaton. In th paper, correct formulae of Jacoban coeffcent of geometrc contrant wth repect to Euler parameter are preented n both Cartean coordnate and relatve generalzed coordnate. Expermental reult how that our propoed formulae make Newton Raphon teratve method converge fater and more table. Y. Lu H.-C. Song J.-H. Yong School of Software, Tnghua Unverty, Bejng , People Republc of Chna e-mal: tyronelau@gmal.com Y. Lu H.-C. Song Department of Computer Scence and Technology, Tnghua Unverty, Bejng , People Republc of Chna Y. Lu H.-C. Song J.-H. Yong Key Laboratory for Informaton Sytem Securty, Mntry of Educaton, Bejng , People Republc of Chna Y. Lu H.-C. Song J.-H. Yong Tnghua Natonal Laboratory for Informaton Scence and Technology, Bejng , People Republc of Chna Keyword Aembly contrant Prmtve contrant Jacoban coeffcent Vrtual rotaton Varatonal method 1 Introducton Aembly modelng play an mportant role n product degn actvte becaue mot product are compoed of a number of part. Intead of degnatng the poton and orentaton of each part, a degner uually pecfe geometrc contrant or knematc contrant between part, whch nvolve olvng thoe contrant to obtan the poture of part [1 3]. Numerou approache to olvng 2D or 3D geometrc contrant problem have been propoed n recent decade. The man dea of them to plt a large geometrc contrant problem nto maller and eaer one ung graph-baed decompong algorthm, rule-baed trategy, algebrac theory, or numercal method. Except for a mall et of ubproblem that can be olved analytcally [4, 5], mot of them wll reort to numercal method, e.g., a Newton-type teratve method or mathematcal optmzaton method [6 11]. To ue a numercal method, the geometrc contrant are converted nto a et of nonlnear equaton frt. Thoe nonlnear equaton are called prmtve contrant or bac contrant. The electon of tate varable can be clafed nto two categore: Cartean coordnate and generalzed coordnate. The Cartean coordnate formulaton yeld a maxmal et of hghly pare equaton, but the form of the equaton ndependent on the topologcal tructure of aembly ytem. On the other hand, recurve generalzed coordnate formulaton can dramatcally reduce the number of contrant equaton and varable, but the form of the equaton vare wth the topologcal tructure of aembly ytem. In both cae, to compute the

3 2226 Int J Adv Manuf Technol : Jacoban coeffcent of contrant equaton, a varatonalvector calculu approach wdely adopted n lterature, whch orgnally preented for dervng the equaton of moton n dynamc [12 16]. The approach ntroduce a concept called vrtual rotaton, wth whch, t convenent to derve the Jacoban coeffcent of prmtve contrant wth repect to the tate varable. However, th concept hould not be appled when Euler parameter are taken a the tate varable, becaue Euler parameter break t prerequte. Otherwe, takng the concept of the vrtual rotaton wll yeld wrong formulae of Jacoban coeffcent, a preented n lterature [6, 9]. In th paper, the prerequte of vrtual rotaton are dcued. The correct Jacoban coeffcent n Cartean coordnate are derved ung varatonal-vector calculu method. Moreover, a recurve formulaton of Jacoban coeffcent of contrant equaton n relatve generalzed coordnate alo derved, whch can be appled to any type of relatve generalzed coordnate, compared wth the retrcted formulaton preented n [6]. The ret of th paper organzed a follow. In Secton 2, we frt generalze prmtve contrant n lterature to a formal repreentaton. The reaon of the wrong formulae preented n lterature dcued. Then the correct formulae of Jacoban coeffcent wth repect to Euler parameter n both Cartean coordnate and recurve generalzed coordnate are derved. In Secton 3, four example are provded to llutrate the correctne and effcency of preented formulae. Fnally, concluon are made n Secton 4. 2 Problem tatement In an aembly ytem, the poture of a rgd body, could be dentfed by r and A,wherer the orgn and A the tranformaton matrx from the local reference frame to the global reference frame. The value n the global reference frame, of a vector v and pont P fxed on body can be obtaned from the followng tranformaton v = A v P = r + v P = r + A v P, where the ymbol wth a upercrpt denote the value of the vector or pont n the local reference frame. There ext an orthogonal condton A A T = I 1 and then the varaton δa atfe δa A T = A δa T 2 A tlde operator on a vector v = [ ] T v x v y v z,form 0 v z v y a kew ymmetrc matrx ṽ = v z 0 v x. From v y v x 0 Eq. 2, thevrtual rotaton δπ defned by δ π = δa A T and δπ = AT δπ. Frt, we generalze prmtve contrant ntroduced n [15] to the followng formal defnton. Defnton 1 A prmtve contrant a calar functon f = fr 1, r 2,, r M, v j1, v j2,, v jn, where r k denote the orgn of body k,andv jk be any vector fxed on body j k. In th paper, we ue th formal defnton to llutrate the dervaton of Jacoban coeffcent of prmtve contrant, ntead of a concrete contrant. The dervaton of Jacoban coeffcent wth repect to both Cartean coordnate and relatve generalzed coordnate are preented. 2.1 Cartean coordnate and Euler parameter A patal mechanm, or aembly, contructed ung Cartean coordnate and Euler parameter, defned n term of the tate varable q = [ ] T r 1 p 1 r n p n, where p = [ e 0 e 1 e 2 e 3 ] T denote Euler parameter of body and atfe the normalzaton condton p T p = 1. Let e = [ e 1 e 2 e 3 ] T, E = [ e, ẽ + e 0 I ],andg = [ e, ẽ + e 0 I ]. A can be expreed a A = E G T.In [15], δπ derved a δπ = 2G δp 3 Ung the vrtual rotaton, Peng et al. [9] derved the varaton of a vector v a δv = δa v = A δ π v = A ṽ δπ = 2A ṽ G δp. Then the varaton of the functon f derved a δf = k = k r k δr k + k r k δr k 2 k δv jk 4 A jk ṽ j k G jk δp jk 5 from whch the Jacoban coeffcent of f are obtaned a f rk = r k, k= 1,,M 6a f pjk = 2 A jk ṽ j k G jk, k= 1,,N, 6b where the partal dervatve r k and are only determned by the form of f telf and eay to compute.

4 Int J Adv Manuf Technol : A ponted out by Yen et al. [17], however, Eq. 6b not the true dervatve of f wth repect to Euler parameter p jk, becaue the defnton of the vrtual rotaton mple the etablhment of Eq. 1 and 2, whch requre p T p = 1and p T δp = 0. It reult n a contradcton. Yen et al. preented a formula of the varaton of a vector v to compute the correct Jacoban coeffcent of prmtve contrant, but h formula complcated. In th paper, ung the varatonalvector calculu method, we derve th varaton n a more conce form a follow. δv = δ A v = δ E G T v = δ e e T + ẽ ẽ + 2e 0 ẽ + e 2 0 I v = δ e e T v + δ ẽ ẽ v + δ 2e0 ẽ v + δ e0 2 v = δe e T v + e δe T v + δẽ ẽ v + ẽ δẽ v + 2δe 0ẽ v Fg. 1 The repreentaton of relatve moton of two bode + 2e 0 δẽ v + 2e 0v δe 0 = e T v δe + e v T δe ẽ v δe ẽ ṽ δe + 2ẽ v δe 0 2e 0 ṽ δe + 2e 0 v δe 0 j = A j j = A j j 10b 10c where = 2 ẽ + e 0 I v δe 0 + e T v I + e v T ẽ v ẽ ṽ 2e 0ṽ δe = K v, p δp, 7 Ku, p = [ 2ẽ + e 0 Iu e T ui + eu T ẽũ ẽu 2e 0 ũ ]. Subttutng Eq. 7 nto Eq. 5 yeld f pjk = K v j v k, p jk, k= 1,,N. 8 jk 2.2 Relatve generalzed coordnate Ung relatve generalzed coordnate, r and A can be repreented recurvely ntead. The relaton of a par of coupled bode, bode and j, depcted n Fg. 1. Vector that locate jont attachment pont n bode and j are denoted by j and j, repectvely. Orthogonal matrce C j, C j,anda j are tranformaton from the jont defnton frame to the body frame on bode and j and from the jont defnton frame on body j to the jont defnton frame on body, repectvely. From Fg. 1,wehavethe followng relaton: r j = r + j + d j j A j = A C j A j CT j and d j = A C j d j 9a 9b 10a where j and j are fxed vector on each body frame eparately. In the equaton above, A j and d j are only the functon of relatve generalzed coordnate q j between bode and j. A mechancal ytem can be repreented by a graph G = GV, E, n whch each node of V repreent a rgd body, and each edge e E repreent the jont between two bode. Both an open-loop and cloed-loop mechancal ytem can be abtracted a a topologcal tree tructure wth addtonal geometrc contrant. Conder any body. There ext a unque path 0 1 from bae body 0 to body. Ung Eq. 9 and 10, r and A can be repreented recurvely by the relatve generalzed coordnate q 0,1, q 1,2,, q 1, [13]. Let q denote the collecton of all relatve generalzed coordnate. To obtan the Jacoban coeffcent of f,the varaton δr and δv hould be derved frt. Let δr and δv have the followng form δr = J r δq δv = J v δq 11a 11b TherecurveformulaeofJ r and Jv were derved ung the vrtual rotaton n [6, 11]. However, mlarly, when a free rotatonal jont nvolved and Euler parameter are choen a the relatve generalzed coordnate, the vrtual rotaton hould not be appled. Otherwe, ung the vrtual rotaton wll yeld erroneou formulae. In th paper, we derve the recurve formulae drectly ntead. Frt, two auxlary coeffcent matrce are ntroduced.

5 2228 Int J Adv Manuf Technol : Defnton 2 Let u be any 3D vector. The auxlary coeffcent matrce J 1, u and J u atfy J 1, uδq 1, = δa 1, u and J uδq = δa u eparately. From the defnton, J 1, only determned by relatve generalzed coordnate q 1, and can be deduced from the jont defnton. For example, f q 1, the Euler parameter, we can obtan t formula from Eq. 7. J 1, uδq 1, = δa 1, u = Ku, q 1,δq 1, 12 TherecurveformulaofJ u derved a follow J uδq = δa u = δ A 1 C 1, A 1, CT, 1 u = δa 1 C 1, A 1, CT, 1 u + A 1 C 1, δa 1, CT, 1 u = J 1 C 1, A 1, CT, 1 δq u + A 1 C 1, J 1, C T, 1 u δq 1, = J 1 C 1, A 1, CT, 1 δq u + [0 A 1 C 1, J 1, C T, 1 u from whch we obtan J u = J 1 C 1, A 1, CT, 1 u + [0 A 1 C 1, J 1, C T, 1 u ] 0 δq ] The varaton δv can be obtaned a δv = δa v = J v δq 14 from whch we obtan J v = J v. 15 The varaton r derved a follow δr = δ r 1 + 1, + d 1,, 1 = δr 1 + δa 1 1, + δ A 1 C 1, d 1, δa, 1 = δr 1 + J 1, δq + J 1 C 1, d + A 1 C 1, δd 1, J, 1 δq 1, δq = δr 1 + J 1, δq + J 1 C 1, d 1, δq + A 1 C 1, D 1, δq 1, J, 1 δq 16 from whch we obtan J r = J r 1 + J 1, + J 1 C 1, d 1, + [ 0A 1 C 1, D 1, 0 ] J, 1, where D 1, the Jacoban coeffcent of d 1, wth repect to local relatve generalzed coordnate q 1,. Subttutng Eq. 14 and 16 nto Eq. 5 yeld δf = k = k r k δr k + k J r r k δq + k k δv jk J v j v k δq jk from whch, the Jacoban matrx of f obtaned a f q = k J r r k + k k J v j v k. 17 jk 2.3 Jacoban coeffcent of prmtve contrant Baed on Haug work [15], ome prmtve geometrc contrant were preented to buld the geometrc contrant lbrary [11]. Mot mportant prmtve contrant are dot- 1 contrant, dot-2 contrant, angle contrant, and dtant contrant. d1 v, v j = v T v j = 0 v, d j = v T d j = v T d j = 0 ang v, v j = v T v j coα 0 = 0 dt d j = d T j d j d 2 0 = 0, where d j = Q j P = r j + A j Q j r A P. 18a 18b 18c 18d The Jacoban coeffcent of thee contrant are derved a follow. For Cartean coordnate, thoe are d1 r = d1 r j = 0 d1 p = v T j K v, p d1 p j = v T v K j, p j 19a 19b 19c r = v T 19d r j = v T p = d T j K v, p v T K p, p p j = v T K p j, p j 19e 19f 19g dt r = 2d T j 19h

6 Int J Adv Manuf Technol : dt r j = 2d T j 19 dt p = 2d T j K p, p 19j dt p j = 2d T j K p j, p j, 19k and for relatve generalzed coordnate, d1 q q dt q = vt j J = dt j J = 2d T j v + v T J j v j v + v T J r j + J j J r j + J j p j J r J p j J r J p p 20a 20b 20c In the next ecton, expermental reult wll be preented to how that the formulae preented n th paper are more feable than thoe propoed n [9]. Algorthm 1: Newton teraton procedure nput : The ntal etmate q 0 the et of equaton output: The olved tatu q q 0 n 0 whle n<nand q <εdo J q q f J nvertble then q q J 1 q end ele /* Ue Penroe-Moore nvere ntead */ q q J + q end n n + 1 end f n == N then return Faled end return Succe 3 Expermental reult In th ecton, we ue the Newton Raphon teraton method lted n Algorthm 1 to evaluate the effectvene of the preented formulae. Four example are ued to llutrate the advantage of our propoed formulae over thoe preented by Peng et al. [9]. Wthout lo of generalty, all the example are two-body contrant ytem wth fxed orgn o that each of them only ha three rotatonal degree of freedom left. Frt, we ue Example 1 to demontrate that the teraton ung Eq. 6b break the aumpton of orthogonalty of the tranformaton matrx, and then ue thee four example to llutrate that our formulae are more applcable. Example 1 Let orgn of bode 1 and 2 be concdent. There are three dot-2 contrant v, P, Q, = 1 3 between the bode. Th contrant ytem well contraned. Wth the parameter gven n Table 1, th contrant ytem ha real oluton. The unknown of the contrant ytem the q = p 2 = [ ] T e 0 e 1 e 2 e 3. The equaton et of the contrant ytem repreented by q T q 1 q = v T Q P,= 1 3 of whch the Jacoban matrx q derved from Eq. 19f and 19g a 2q T Q q q = 1 P 1 T K v 1, q v T 1 K P 1, q Q 2 P 2 T K v 2, q v T 2 K P 2, q 21 Q 3 P 3 T K v 3, q v T 3 K P 3, q whle Eq. 6b gve 2q T 2 v T 1 P 1 Q 1 P 1 T Aṽ 1 G ˆ q q = 2 v T 2 P 2. 2 P 2 T Aṽ 2 G 22 2 v T 3 P 3 Q 3 P 3 T Aṽ 3 G Gven an ntal etmate q 0 = [ ] T, whch atfe the normalzaton contrant, we can get A = and AA T = I, Subttutng A nto Eq. 22 yeld ˆ q0 =

7 2230 Int J Adv Manuf Technol : Table 1 The parameter ued n example Example Parameter v 1 = 1 5, v 2 = , v 3 = , P 1 = 4, P 2 = 0, P 3 = 3, Q 1 = 5, Q 2 = 1, Q 3 = P 1 = 0, P 2 = 0, P 3 = 1, Q 1 = 0, Q 2 = 1, Q 3 = 0, d 1 = 1, d 2 = 6, d 3 = u 1 = 0, u 2 = 1, v 1 = 0, v 2 = 1, u 3 = 1 3, P 1 = 1, Q 1 = u 1 = 0, u 2 = 1, v 1 = 0, v 2 = 1, P 1 = 1, Q 1 = 1, α 1 = π 6, α 2 = π 3, d 1 = From Algorthm 1, we can get the next teraton pont q 1 = [ ] T, and the tranformaton matrx A = whch break Eq. 1. Contnung th procedure, we wll fnd that the teraton fal to converge. On the other hand, f we adopt Eq. 21, the teraton converge n about 10 tme gven a tolerance To tudy the convergency generally, many ntal etmate are generated to tet whether Algorthm 1 converge, and f t doe, n how many teraton. Our tet are Table 2 The mulaton reult Experment Etmate regon Jacoban formulae Convergent cae Average teraton Exam. 1, hyper box Propoed 10, body 2 movable Peng 9, hyper phere Propoed 10, Peng 9, Exam. 1, hyper box Propoed 3, body 1 movable Peng hyper phere Propoed 2, Peng Exam. 2 hyper box Propoed 8, Peng hyper phere Propoed 6, Peng 1, Exam. 3 hyper box Propoed 1, Peng 1, hyper phere Propoed 1, Peng 1, Exam. 4 hyper box Propoed 8, Peng 7, hyper phere Propoed 8, Peng 8,

8 Int J Adv Manuf Technol : performed a follow. For each example, we generate 10, 000 ntal etmate unformly at random from an etmate regon to tet whether Algorthm 1 converge, ung our and compared formulae eparately. We count the convergent cae and calculate the average teraton needed for each ucceful cae. The two etmate regon to concern are 1. {e 0,e 1,e 2,e 3 e 1} denoted a hyper box 2. {q 0 q T 0 q 0 = 1} denoted a hyper phere The dfference between thee two regon that the latter the et where the ntal etmate atfe the normalzaton condton. Other ued example are lted a follow: Example 2 The contrant ytem cont of three dtant contrant dt P, Q,d, = 1 3. Example 3 The contrant ytem cont of two dot-1 contrant d1 u, v, = 1, 2 and one dot-2 contrant u 3, P 1, Q 1. Example 4 The contrant ytem cont of two angle contrant ang u, v,α, = 1, 2 and one dtant contrant dt P 1, Q 1,d 1. The parameter ued n the example are gven n Table 1. The mulaton reult are lted n Table 2. The reult how that the teraton method adoptng our preented formulae a Jacoban coeffcent are more table. A depcted n Table 2, there are more convergent cae n general, ung propoed formulae. Moreover, our propoed formulae wll make the teraton method converge fater n fewer teraton. The expermental reult demontrate the feablty of the method preented n th paper. 4 Concluon In th paper, the erroneou dervaton of Jacoban coeffcent of prmtve contrant equaton wth repect to Euler parameter preented n lterature dcued, and correct formulae are derved ung varatonal-vector calculu method. A recurve formulaton of Jacoban coeffcent of prmtve contrant n relatve generalzed coordnate alo propoed. Compared wth the formulae preented n lterature, th formulaton can be appled to all type of jont varable. Expermental reult are preented to llutrate the correctne and computatonal effcency of the formulae preented n th paper. Acknowledgment The reearch wa upported by Chnee 973 Program 2010CB and Chnee 863 Program 2012AA The frt author wa upported by the NSFC , The econd author wa upported by the NSFC The lat author wa upported by the NSFC Reference 1. Km SH, Lee K 1989 An aembly modellng ytem for dynamc and knematc analy. Comput Aded De 21: Anantha R, Kramer GA, Crawford RH 1996 Aembly modelng by geometrc contrant atfacton. Comput Aded De 28: Km JS, Km KS, Lee JY, Jeong JH 2005 Generaton of aembly model from knematc contrant. Int J Adv Manuf Technol 261 2: Kramer GA 1992 A geometrc contrant engne. Artf Intell 58: Km J, Km K, Cho K, Lee JY 2000 Solvng 3D geometrc contrant for aembly modellng. Int J Adv Manuf Technol 16: Zou H, Abdel-Malek K, Wang J 1996 Computer-aded degn ung the method of cut-jont knematc contrant. Comput Aded De 28: L YT, Hu SM, Sun JG 2002 A contructve approach to olvng 3-D geometrc contrant ytem ung dependence analy. Comput Aded De 34: Km JS, Km KS, Lee JY, Jung HB 2004 Solvng 3D geometrc contrant for cloed-loop aemble. Int J Adv Manuf Technol 23: Peng XB, Lee KW, Chen LP 2006 A geometrc contrant olver for 3-D aembly modelng. Int J Adv Manuf Technol 28: Sh ZL, Chen LP 2007 An angular contrant olvng approach for aembly modelng baed on phercal geometry. Int J Adv Manuf Technol 32: Xa HJ, Wang BX, Chen LP, Huang ZD D geometrc contrant olvng ung the method of knematc analy. Int J Adv Manuf Technol 35: Haug EJ, McCullough MK 1986 A varatonal-vector calculu approach to machne dynamc. J Mech Tranm Autom De 1081: Bae DS, Haug EJ 1987 A recurve formulaton for contraned mechancal ytem dynamc: Part I. Open loop ytem. Mech Struct Mach 15: Bae DS, Haug EJ 1987 A recurve formulaton for contraned mechancal ytem dynamc: Part II. Cloed loop ytem. Mech Struct Mach 15: Haug EJ 1989 Computer aded knematc and dynamc of mechancal ytem: bac method. Allyn and Bacon, Boton 16. Bae DS, Han JM, Yoo HH 1999 A generalzed recurve formulaton for contraned mechancal ytem dynamc. Mech Struct Mach 273: Yen J, Chou CC 1993 Automatc generaton and numercal ntegraton of dfferental-algebrac equaton of multbody dynamc. Comput Method Appl Mech Engrg 1043: