EFFICIENT COMPUTATION OF THE GENERALIZED INERTIAL TENSOR OF ROBOTS BY USING THE GIBBS- APPELL EQUATIONS

Transcription

1 EFFICIEN CMPUAIN F HE ENERALIZED INERIAL ENSR F RBS BY USIN HE IBBS- APPELL EQUAINS Povenzano S. (*) Mata V.(**) Ceccaell M.(***) and Suñe J.L. (**) (*) Escuela de Ingeneía Mecánca Unvesdad de Los Andes (VENEZUELA) (**) Depatamento de Ingeneía Mecánca y de Mateales Unvesdad Poltécnca de Valenca (SPAIN) (***) Dpatmento d Meccanca Stuttue Ambente e etoo Unvestà d Cassno (IALY) Abstact In ths pape a method fo computng the genealzed netal tenso of manpulato obots s descbed as based on the bbs-appell euatons. he method pesents the fomulaton of the coeffcents of the mentoned tenso by means of the Hessan of the bbs functon. An effcent ecusve algothm s poposed wth a complexty of (n 2 ) ode whch s used to poduce a pogam n FRRAN. A numecal example smulates the movement of a PUMA obot. he esults obtaned fom the smulaton ae compaed wth those obtaned by usng othe nown methods.. - Intoducton he dect dynamc poblem o smulaton poblem of obots s a pocess that nvolves the computng of acceleatons veloctes and oentatons o ont postons fo a manpulato when the foces exeted by the actuatos ae gven. he smulaton poblem fo mechancal systems conssts of two pats: the fst one eues the calculaton of ont acceleatons fom the movement euatons; and the second one nvolves the ntegaton of those euatons. he ont acceleatons can be obtaned by two dffeent appoaches. In the fst one a system of dffeental euatons s poposed obtaned and solved. In the second appoach the acceleatons ae calculated ecusvely by popagatng the movement and constant foces thoughout the mechansm. he algothms deved fom the fst appoach have usually a computatonal complexty of (n 3 ) ode manly due to the fact that a lnea system must be solved. Papes [] and [2] ae

2 examples of ths appoach. n the othe hand the algothms deved usng the second appoach have a computatonal complexty of (n) ode although the coeffcents that appeas ae lage (fo example [3-6]). hese algothms ae supeo than the fst ones but only fo systems wth moe than 9 ln as shown n [7]. he best nown appoach s the atculated body (ABM) [4]. he algothms based on the fst appoach follow the pocedue that has been developed by Wale y n [] whch eues the genealzed netal tenso and the so-called bas vecto. In Wale and n s wo they obtan the bas vecto by means of an algothm that solves the nvese dynamc poblem n whch the ont acceleatons became null. he claculaton of the genealzed netal tenso s then the man poblem fo the applcaton of ths appoach. Wale and n use the composed gd body method (CRBM) whch s based on the Newton-Eule euatons and has a complexty of (n 2 ). An altenatve method has been developed by Angeles and Ma [2] who use the natual othogonal complement to obtan an effcent algothm but of (n 3 ) ode. hese methods have been studed and mpoved ove tme (see fo example [7] [8]) even by usng the spatal algeba as n [9]. It has been demonstated that the methods ABM y CRBM ae dffeent foms of solvng the same lnea system. hs fact s shown n seveal papes such as [0] and []. In ths pape a new algothm s poposed as based on the bbs-appell euatons n ageement wth [2] n ode to compute the genealzed netal tenso. he use of the bbs-appell euatons n the dynamcs of multbody systems has been lmted n the last twenty fve yeas to few examples that followed manly a closed fom fomulaton (see [3]) so that a geat numbe of athmetc opeatons wee eued. he fst ecusve development of (n 2 ) ode was poposed to solve the nvese dynamc poblem by Desoye and Lugne [4] who used the Jacoban matx of the manpulato n ode to avod the explct development of patal devatves. Late Rudas and oth [5] pesented a ecusve algothm of (n) ode lewse to solve the nvese poblem. hs algothm s based on the mnmsaton of the bbs functon by means of Lagange s multples. Recently Mata et al. [6] have solved the nvese dynamc poblem fo obotc manpulato and ntoduced an (n) effcent ecusve algothm whch taes advantage of the possbltes that the bbs-appell euatons offe to compute the genealzed foces on obotc manpulatos. In the next secton we wll stat fom the gd body s bbs functon to ave to an expesson that pemts to develop of effcent ecusve elatons whch ae useful to compute the tems of the genealzed netal tenso. In the thd secton a bbs-appell Hessan (AH) algothm s poposed as based on that expesson. In the foth secton an example s shown n whch the esults obtaned fom smulatng the movement of a PUMA obot ae compaed wth those obtaned fom ABM CRBM and the AH methods. 2.- enealzed netal tenso by usng the Hessan of the bbs functon he bbs functon fo a gd body that s pat of a mechancal system can be expessed as [2] 2

3 m 2 ( ) + ( ) I + ( ) ( I ) whee m s the mass of -th ln 2 () the acceleaton of the cente of mass of -th ln expessed n the local efeence system and the angula velocty and acceleaton of -th ln both expessed n the local efeence system and I the neta tenso of ln -th expessed n ts own efeence system. n the othe hand the whole bbs functon fo a mechancal system wth n bodes s gven by n Consdeng the expessons () and (2) the tems of D whch s the genealzed netal tenso wll be gven by the Hessan matx of the bbs functon n the fom 2 D whee s the genealzed acceleaton of -th ont. Fom the pevous expesson a smple algothm of (n 3 ) ode can be obtaned fo to the computaton of D although s possble to educe t to a ecusve algothm of (n 2 ) ode. A pocedue can be used to fnd the expesson that pemts to calculate the tems of the genealzed netal tenso though successve patal devatves of the bbs functon. It can be futhe smplfed f the followng expessons ae taen nto account when the applcaton of the Denavt-Hatenbeg s modfed notaton s used [2] z 0 o o R n whch z s the unt vecto along the Z axs o the acceleaton of the ogn of the (- )-th efeence system R the otaton matx between two adacent efeence systems s the vecto fom the ogn of the (-)-th efeence system to the ogn of the -th efeence system s the vecto fom the ogn of the -th efeence system to the cente (2) (3) 3

4 4 of mass of the -th ln. he supescpt denotes the efeence system on whch the vectos ae expessed. hus the expesson fo the tems of the genealzed netal tenso loos le + n m D I (4) he utlsaton of ths last expesson can elaboate a computatonally effcent algothm applcable to manpulatos but t needs the study of the ecuent elatons that among ts elements snce the dect applcaton of the expesson would poduce an algothm wth a hgh numbe of opeatons. he followng expessons ae useful fo a ecusve calculaton of the expesson (4): R In addton eplacng the vectoal poduct by the poduct of an antsymetc tenso and a vecto. he followng expesson can be obtaned Applyng the popetes of the vectoal poduct the euaton Eoe. L'agomento paameto è sconoscuto. can be wtten as + n m D I By usng the antsymetc tenso as pevously defned the expesson taes the fom ( ) n m m D I R (5) 3.- AH Algothm A ecusve algothm of (n 2 ) ode can be fomulated as based on the expesson (5) n ode to compute the genealzed neta tenso fo a obotc manpulato. In patcula the computaton can be povded by means of a step by step pocedue as n the followng. Step he compound mass the compound netal tenso and othe uanttes that do not vay wth the tme can be calculated off-lne by means of the followng tems once M n m n s gven Fo to n do

5 M m + M + 0M 0A (6) γ m + M 0M 0A (7) + + Fo to n do I 0M 0A (8) I m I M E 0M 0A (9) Step 2 Vectos can be computed by Fo to n Fo + to n do R 8M 4A (0) + + Fo n 2 to Fo n to + 2 do M 3A () Step 3 Vectos.can be evaluated by the ecusve computatons n the fom Fo to n set [ 0 0 ] Fo 2 to n set R (3) (23) (33) [ R R ] 0M 0A (2) 0M 0A (3) Fo to n Fo + 2 to n do R 8M 4A (4) 5

6 Step 4 Vectos φ y φ (the poduct m can be obtaned off-lne) can be obtaned once n n φ s gven by means of the computaton n m n n n Fo n to do + φ+ R+ φ + 8M 4A (5) φ γ + φ 0M 3A (6) + Step 5 Fo the computaton of ensos ø. he esemblance tansfomaton s calculated n an + + n n effcent way by dong the decomposton of the poduct R+ ø + R once ø n In s gven and the tems ae calculated as Fo n to 2 do ø ( + + ) R+ ø + R E + φ + φ M 40A (7) (33) It s to note that Fo only the element ø s necessay 3M A (8) Step 6 he elements of pncpal dagonal of the genealzed netal tenso can be computed by Fo to n do D Step 7 ø 0M 0A (9) Computaton of the est of elements of the genealzed netal tenso Fo n to 2 Fo to do D [ ø ] φ 7M 6A (20) 6

7 able.- Compason of complextes Authos Method Poducts M (n 6) Addtons A(n 6) Wale and n CRBM 2n 2 +56n 27 (74) 7n 2 +67n 53 (60) Angeles and Ma Nat. tog. Comp. n 3 +7n 2 2n+8 (70) n 3 +4n 2 6n+5 (629) hs wo AH.5n n 49 (482) 8.5n n 69 (426) nce the genealzed neta tenso s detemned a numecal smulaton of dynamc behavou of a multbody system can be staghtfowad obtaned. 4.- Example In ode to execute an example useful fo compason and valdaton a smulaton pogam has been wtten n FRRAN as based on the methodology poposed by Wale and n [] n whch fo the genealzed netal tenso that s calculated by usng the poposed algothm. In addton the bas vecto s calculated by usng a modfed veson of the algothm that solves the nvese dynamc poblem by usng the bbs-appell euatons poposed n [6]. he pogam has been wtten n double pecson and s executed on a compute PC Pentum II 400 MHz. he auss-jodan elmnaton method has been used to solve the lnea system and the system of dffeental euatons s ntegate by the Runge-Kutta s fve ode technue. In the ntegaton pocess the toleance s 0-6 and the tme nteval of 0. s. In ths example we smulate the movement of the PUMA obot whose temnal element descbes a staght taectoy wth constant oentaton and the velocty of 0. m/s. he 0 smulaton stats fom the poston [ ] the oentaton (Z Y 0 CP t 0s Z) of the local system fxed on the ln 6 s [ 60 90] 45 deg. Afte 5 seconds the movement lasts the temnal element eaches the poston gven by Fgue shows the obot n the ntal poston and the 0 CP t 5s [ ] planned taectoy. he nvese dynamc poblem has been solved fo each nstant by usng as nput data the esults of the esoluton of the nvese nematc poblem though the pocedue that s outlned n the flowchatof Fgue 2. he supescpt "*" denotes the genealzed coodnates veloctes and acceleatons obtaned fom the numecal ntegaton of the poposed fomulaton. In ode to valdate and compae the esults smulaton codes ae wtten n FRRAN as based on the composed gd body methods (CRBM) and of atculated body (ABM). hose codes have been executed unde dentcal condtons also use fo the code that s based on the algothm AH. he dffeences among the obtaned esults fom the thee compaed methods ae subtle so that s possble to obseve dffeences n genealzed coodnates veloctes and acceleatons fom the tenth decmal. In the othe hand f the esults ae expessed egadng to poston velocty and acceleaton of the end-effecto the dffeences can be obseved moe clea due to the accumulaton eos poduced n the dffeent onts that compose the studed system. In the able 2 the maxmum eos wth espect to the 7

8 pescbed taectoy and the dffeence aveage ae depcted whch tae place n the modules of the vecto poston velocty and acceleaton of the element temnal obtaned fom the smulaton wth each one of the methods. 0 0 CP 0 0 CP t 0s t 5.0 s 0 0 CP CP 0 Invese Knematc Poblem t 0 s s t 0s s t 0s s Invese Dynamc Poblem τ t0s s Fgue.- Intal poston of the obot and taectoy planned. Dect Dynamc Pob. * * * t 0s s t 0 s s t 0s s Fgue 2.- Flow chat of the pocedue fo the numecal smulaton able 2.- Maxmum Eos and pocessng tme n the smulaton fo each method. CRBM ABM AH 0 0 CP 0 CP 0 CP Max Pom Max Pom Max Pom Pocessng tme (s.)

9 5.- Concluson In ths pape the bbs-appell euatons have been used fo the fomulaton of a new ecusve algothm fo the computaton of the genealzed netal tenso. he algothm s of (n 2 ) ode and because of the athmetc numbe of nvolved opeatons t s computatonally effcent as shown n the able. In the algothm the tems of the genealzed netal tenso ae obtaned by means of the Hessan matx of the bbs functon. hs appoach taes the advantage of the bbs euatons fo a novel sutable fomulaton of the genealzed neta tenso of mult-body systems patculaly obotc manpulatos. A numecal example has been caed out n a PUMA obot usng the poposed algothm (AH) and the algothms bette nown of the lteatue (CRBM and ABM). he esults of these smulatons ae shown n the able 2 whee t can be obseved that the method AH poduces eos smalle than the CRBM method but a lttle bt geate than the ABM whch s accodng to seveal authos ( see fo example [] and [7]) the method that has bette numecal esponse. In addton fom able 2 the pocessng tme of the AH method s the smallest one snce the poposed algothm has smalle computatonal complexty wth espect to tadtonal the algothms deved fom the CRBM and ABM methods. 6.- Refeences [] M. W. Wale D.E. n Effcent Dynamc Compute Smulaton of Robotc Mechansms Jounal of Dynamc Systems Measuement and Contol vol 04 pp [2] J. Angeles. Ma Dynamc Smulaton of n-axs Seal Robotc Manpulatos Usng a Natual thogonal Complement he Intenatonal Jounal of Robotc Reseach vol. 7 no. 5 pp [3] W. W. Amstong Recusve soluton to the euatons of moton of an n-ln manpulato Poc. Ffth Wold Congess on heoy of Machnes and Mechansms Monteal 979. [4] R. Feathestone he Calculaton of Robot Dynamcs usng Atculated-Body Inetas Int. J. Robotcs Reseach vol. 2 no. pp [5]. Rodguez Kalman Flteng Smoothng and Recusve Robot Am Fowad and Invese Dynamcs IEEE Jounal of Robotcs and Automaton vol. RA-3 no. 6 pp [6] S. K. Saha A decomposton of the manpulato neta matx IEEE ans. on Robotcs and Automaton vol. 3 no [7] R. Feathestone D. n Robot Dynamcs: Euatons and Algothms Poc. f the 2000 IEEE Int. Conf. n Robotcs Automaton pp San Fancsco Abl [8] C. A. Balafouts R. V. Patel Dynamc Analyss of Robot Manpulatos: A Catesan enso Appoach Kluwe Academc Pess Boston 99. [9]. Rodguez A. Jan K. Keutz-Delgado A Spatal peato Algeba fo Manpulato Modellng and Contol Int. J. Robotcs Reseach vol. 0 no. 4 pp [0] D. K. Pa U. M. Asche P.. Ky Fowad dynamcs algothms fo multbody chans and contact Poc IEEE Int. Conf. on Robotcs Automaton San Fancsco Apl

10 [] U. M. Asche D. K. Pa B. P. Cloute Fowad Dynamcs Elmnaton Methods And Fomulaton Stffness n Robot Smulaton he Intenatonal Jounal of Robotc Reseach vol. 6 no. 6 pp [2] L. Pas A eatse on Analytcal Dynamcs x Bow Pess Connectcut 972. [3] M. Vuobatovc y N. Kcans Real me Dynamcs of Manpulaton Robots. Spnge-Velag Beln 985. [4] K. Desoye P. Lugne H. Spnge Dynamc effects of actve elements n manpulatos and the nfluence upon the contollng dves IUAM/IFFoMM Symposum pp Udne 985. [5] I. Rudas A. oth Effcent Recusve Algothm fo Invese Dynamcs Mechatoncs vol 3 no 2 pp [6] V. Mata S. Povenzano J. I. Cuadado F. Valeo An (n) Algothm fo Solvng the Invese Dynamc Poblem n Robots by Usng he bbs-appell Fomulaton Poc. of enth Wold Congess IFoMM vol. 3 pp ulu June