Dynamics of a Spatial Multibody System using Equimomental System of Point-masses

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1 Dynamcs of a Spatal Multbody System usng Equmomental System of Pont-masses Vnay Gupta Dept. of Mechancal Engg. Indan Insttute of echnology Delh New Delh, Inda vnayguptaec@gmal.com Hmanshu Chaudhary Dept. of Mechancal Engg. MNI Japur, Inda hchaudhary@mnt.ac.n Subr K. Saha Dept. of Mechancal Engg. Indan Insttute of echnology Delh New Delh, Inda saha@mech.td.ac.n Abstract hs paper presents dynamc modellng of a spatal multbody system usng e equmomental system of pont-masses whch has several advantages, e.g., one needs to wrte only e translatonal equatons of moton wout e necessty of wrtng e rotatonal equatons as e pont masses have no dmenson. For s, a rgd body s represented as a set of rgdly connected seven pont-masses. Accordngly, e veloctes of e pont-masses are derved as a lnear transformaton of e jont-rates resultng nto a set of ree decoupled matrces called e Decoupled Natural Orogonal Complement (DeNOC) matrces for e pontmasses. he matrces are en used to form a mnmum set of constraned equatons of moton from e uncoupled Newton s equatons of lnear moton. he meodology s llustrated usng a spatal sold pendulum. Keywords Dynamcs, spatal multbody system, equmomental system, DeNOC. I. INRODUCION Dealng w e dynamcs of a spatal multbody system s always a challengng task. Dynamc equatons of moton of e multbody system can be wrtten usng varous coordnate systems. Absolute coordnate system represents e confguraton of a body relatve to e global reference frame. hs results n a large number of equatons of moton []. In jont coordnate system, e generalzed coordnates are e relatve coordnates between e bodes connected by a knematc par or jont []. On e oer hand, an equmomental system of pont masses s e one whch conssts of several pont masses whose total mass, overall mass-center locaton and e moment of nerta are same as at of e orgnal rgd body [3-4]. Such representaton has several advantages. For example, one needs to deal w e translatonal equatons of moton only, as e pont masses have no dmenson. Besdes, t s more convenent to perform optmum desgn of e shapes of e bodes formng e overall multbody system, and ts balancng. hese advantages have motvated e researchers to use e concept of Natural coordnates, as presented n [5] and oers. In [6] pont coordnates replaced e rgd body and e equatons of moton were derved n jont coordnates usng e velocty transformaton matrx. In [7-8] pont coordnates s used to represent e rgd body and lnear and angular momentum concept appled for dynamcal formulaton. he pont mass representaton was also used n [9-], where balancng of several mechansms was carred out. hs paper presents e modellng of a spatal multbody system usng e concept of equmomental system of pont-masses and e correspondng Decoupled Natural Orogonal Complement (DeNOC) matrces. A set of seven pont-masses, as proposed n [], was used to defne a rgd body movng n e ree-dmensonal Cartesan space. he preference of seven pont-masses over typcally used four pont-masses [3] s manly due to e presence of lnear set of algebrac equatons to fnd e pont-masses []. he pont-mass veloctes were en expressed as a lnear transformaton of e jont-rates. he assocated matrx s called e DeNOC matrces for e pont-mass system, smlar to e one presented n [-3] for a seral-chan system. Furer e equatons of moton were derved usng e Newton s equatons of lnear moton only. he meodology s shown by applyng t to a spatal sold pendulum. A smlar meodology was presented n [4] for a planar system. he paper s organzed as follows: Secton II ntroduces e concept of pont-mass system for a rgdbody n a spatal moton, whereas Secton III derves e DeNOC matrces for e pont-mass system. Secton IV presents dynamc modelng and Secton V llustrates e modelng usng a spatal pendulum. Fnally, Secton 6 concludes e paper. II. EQUIMOMENAL SYSEM OF SEVEN POIN-MASSES Consder a rgd body or lnk, denoted as lnk, movng n ree-dmensonal Cartesan space. he rgd lnk s dynamcally represented as a set of rgdly connected seven pont-masses, as proposed n [0-]. It s shown n e Fg.. o avod e concdence of any two ponts, e pont-masses m are located at e vertces of a parallelepped, whose center s located at e orgn O + whch has sdes hx, hy,hz at are parallel to e axes of e body-fxed frame O + X+ Y + Z+. he two systems of rgd-body and seven pont-mass system are equmomental f e followng condtons are satsfed: m + m + m + m + m + m + m = m () Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec

2 ( m + m m m m + m + m ) h = m x () x ( m + m + m + m m m m ) h = m y (3) y ( m m m + m + m + m m ) h = m z (4) z ( m + m m m + m m m ) h h = I (5) x y, xy ( m m m + m m m + m ) h h = I (6) y z, yz ( m m + m m m + m m ) h h = I (7) z x, zx 7 m ( hy + hz ) = I, xx (8) j= 7 m ( hz + hx ) = I, yy (9) j= 7 m ( hx + hy ) = I, zz (0) j= In (-0), m s mass, ( x, y, z ) s mass-center locaton, ( I,, I,, I, ) are e moment of nerta, and, xy, yz, zx xx yy zz ( I, I, I ) are e product of nerta of e lnk at hand. he mass-center and e nerta tensor are referred n e body-fxed frame denoted as O + X+ Y + Z+. Equaton n () ensures e mass of e equmomental system s same as at of e rgd body, whereas ose n (-4) represent e same mass-center locatons, and n (5-0) ensure e same nerta tensors about pont O +. Note at (-0) are nonlnear n parameters, m, hx, hy and h z. he parameters hx, hy and h z were found separately usng (8-0), as proposed n [0-]. hs s done n e followng way: x, xx, yy, zz h = ( I + I + I ) / ( m ) () y, xx, yy, zz h = ( I I + I ) / ( m ) () z, xx, yy, zz h = ( I + I I ) / ( m ) (3) Note at e sum of any two moment of nertas to be greater an e rd one [3]. Hence, hx, hy and hz wll never have magnary values. hen (-7) becomes lnear n m,..., m 7 and solve for em. hs wll be carred out n Secton V. III. DENOC MARICES FOR HE POIN-MASS SYSEM Consder e rgd lnk n spatal moton. he lnk s represented as a set of rgdly connected seven pontmasses shown n Fg.. he veloctes of e pont-masses of e lnk s derved n general form from e velocty of e orgn O of e lnk as whereω and v = v + ω d (4-0) v = v + ω d 7 7 v are e angular velocty and lnear velocty of e orgn O of e lnk, respectvely, whereas e vector d = a + r, for j =,,7, denotes e poston of e orgn O. Vector j pont-mass of e body from ts a s e lnk leng vector of e O + body,.e., e vector representng e poston of O, whereas vector r, j =,...,7, s e poston of e O + from pont-mass from. For a system of seven pont-masses, er veloctes are gven by (4-0), whch can be expressed n a compact form as j v ɶ = Dt () where vɶ s e -dmensonal vector of pont-mass veloctes, and D s e 6 pont-mass matrx defned as d D () d 7 Fg.. Seven pont-mass system In (), d, for j =,...,7, s e 3 3 cross-product tensor assocated w e vector d,.e., ( d ) x = d x, for any 3-dmensonal Cartesan vector x. Moreover, e term s e 3 3 dentty matrx, and t s defned as e twst of e body [, 3]. For a rgd Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec

3 body movng n e ree-dmensonal Cartesan space, e 6-dmensonal twst s defned as ω t v (3) where ω and v are e 3-dmensonal vectors of angular velocty and lnear velocty of e orgn of e body,.e., O, respectvely. If e rgd bodes are coupled w e help of a sngle-degree-of-freedom (DoF) jonts, say, a revolute and prsmatc, e twsts of all rgd bodes n e system represented as a generalzed twst,.e., t t t n can be wrtten as t = Nθ ɺ, where θ ɺ ɺ θ ɺ θ n (4) and e matrx N s referred to as e Natural Orogonal Complement or NOC of e velocty constrant matrx [3]. However, f e bodes or e lnks are connected w a jont havng more an one-dof jont, say, a 3-DoF sphercal jont, en e scalar ɺ θ assocated w e jont needs to be replaced w e 3-DoF jont-rate vector. Now, f a multbody system has seral-chan w n rgd bodes, e veloctes of ts equmomental pont-masses can be expressed as v ɶ = Dt (5) where vɶ s e n-dmensonal vector, t s e 6ndmensonal vector of generalzed twst, and D s n 6n matrx defned as [ ] dag.,..., n D D D (6) Combnng (4-6), one can wrte e pont-mass veloctes vɶ, n terms of e jont rates,,e., θ ɺ, as vɶ = Nθ ɶ ɺ, where Nɶ DN and N NlN d (7) he matrces Nl and Nd for one-dof jonts couplng e bodes are gven n [3], whch are gven by O O p 0 0 A O l 0 p 0 N N d (8) A A 0 0 p n n n Where O s e matrx of zeros whose dmenson s compatble accordng to e matrx where t appears. For example, O of N l s e 6 6 matrx of zeros. Smlarly, n N d of (8), 0 represents e 6-dmensonal vector of zeros. Moreover, denotes e 3 3 dentty matrx, and e 6 6 matrces of N l,.e., A,, called twst propagaton matrx. Fnally, e vectors of N d,.e., p, are e jontmoton propagaton vector. he expressons for e 6 6 matrx, A,, and e 6-dmensonal vector p are gven by A, Ο e p a, 0 (9) n whch a, = a s e vector assocated to e skewsymmetrc matrx a,. Note here at f e jonts are havng more DoF, e jont-moton propagaton vector becomes a matrx whose columns are correspondng to e DoF of e jont at hand. For a 3-DoF sphercal jont, llustraton wll be gven n Secton V. IV. DYNAMIC MODELLING he equatons of moton for e pont-mass system were derved usng Newton s equatons of lnear moton and e correspondng DeNOC matrces derved n Secton III. he unconstraned Newton s equatons of moton for e j pont-mass of e body are gven by m v ɺ = f, for j =,...,7 (30) where m s e mass of e j pont-mass n e whereas, body, vɺ s e acceleraton of pont-mass, and f s e force ncludng gravty actng on e j pont mass of e body. Equaton n (30) can be wrtten for an n-lnk system as Mv ɶ ɺɶ = fɶ (3) In (3), n n matrx M ɶ and e n-dmensonal vectors vɺ ɶ and fɶ are defned as follows: [ ] M ɶ dag. m... m,..., m... m (3a) 7 n n7 vɺɶ vɺ vɺ vɺ vɺ (3b) =... 7,..., n... n7 fɶ f f f f =... 7,..., n... n7 (3c) Now, upon pre-multplcaton of e transpose of N ɶ of (7) to (3) gves e mnmal set of constraned equatons of moton,.e., ɶ ɶ ɺɶ = ɶ ɶ (33) N Mv N f where vɺ ɶ = Nθ ɶ ɺɺ + Nθ ɶ ɺ ɺ. Equatons n (33) lead to e set of ndependent equatons of moton n generalzed coordnates gven by Iθ ɺɺ + Cθɺ = τ (34) n whch e I s n n matrces, s called e Generalzed Inerta Matrx (GIM) and Cs e matrx contanng convectve nerta terms. Moreover,τ s n-dmensonal vector of torques. he expressons of I, C and τ are gven by Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec

4 Fg.. Spatal pendulum w ts seven pont model. Fg. 3. Spatal pendulum at zero Euler angles I Nɶ MN ɶ ɶ C Nɶ MN ɶ ɶ ɺ τ = Nɶ fɶ (35) Note at e expressons n (35) provde e same set of fnal expressons obtaned usng e Newton-Euler uncoupled equatons of moton and e correspondng DeNOC matrces, as derved n [,3]. V. ILLUSRAION: SPAIAL PENDULUM Consder a spatal pendulum shown n Fg.. he rgd body of e pendulum s coupled to e fxed-base by a sphercal jont whch has 3-DoF. he pendulum s represented by e set of rgdly connected seven pontmasses. he pont-masses are located at e vertces of a parallelepped whose center s located at orgn O of e body-fxed frame, as shown n Fg.. For e jont moton of e 3-DOF sphercal jont, Euler Angles (EA) can be used. Out of possble EA sets to represent a sphercal jont, e ZYZ set was chosen for e purpose of dynamc modellng of e spatal pendulum usng e system of pont-masses. For e ZYZ set, e rotaton matrx requred for several coordnate transformaton of matrces and vector s gven from [3] as CφCθ Cϕ SφSϕ CφCθ Sϕ SφCϕ Cφ Sθ Q SφCθ Cϕ + CφSϕ SφCθ Sϕ + CφCϕ SφSθ (36) SθCϕ Sθ Sϕ Cθ where S and C stand for e Sne and Cosne, whereas φ, θ, ϕ are e ZYZ Euler angle set. Fg. 3 shows e pendulum poston for zero EA set. he coordnates of e pont-masses or e component of vector r j, for j =,...,7, calculated usng (-3) n e body-fxed frame are gven n able. he mass of e pont-masses were calculated usng (- 7) for e followng nput parameters of e rgd body: m =.5 kg, a =m, dameter of lnk, d =.0m, mass center locaton, x = a /, y = 0, z = 0, and e moment of nerta and e product of nerta, I xx = md / 8, I = I = ( md /6) + ( ma / 3), I = I = I = 0 yy zz xy yz zx kg-m. Note at e mass-center and e nerta tensor were referred n e body-fxed frame O X Y Z. he pont-masses were en calculated as, m = 0.68kg m = 0.543kg m3 =.7kg m4 = 0kg m5 =.7kg m6 = 0.543kg m7 = 0.68kg. A. General Inerta Matrx he general nerta matrx was derved usng e expresson gven n (35) as, I Nɶ MN ɶ ɶ, where e mass matrx for e seven pont-mass system s gven as, M ɶ dag. m,..., m and Nɶ DN N, where e [ ] 7 l d ABLE I. COMPONEN OF VECOR rj IN BODY-FIXED FRAME, I.E., O X Y Z r r r 3 r 4 r 5 r 6 r 7 Along X Along Y Along Z h h - h - h - h h h x x x x x x x h h h h - h - h - h y y y y y y y h - h - h h h h - h z z z z z z z Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec

5 6 pont-mass matrx D, e 6 6 matrx N l, and e 6 3 matrx N d for e 3-DoF sphercal jont, are gven by d O L D N N O O d7 l d (37) where e 3-dmensonal vectors [ d j ], for j =,,7 n body-fxed frame s gven as [ d ] a + hx a + hy a + hz [ d ] a + hx a + hy a hz [ d 3] a hx a + hy a hz [ d 4] a hx a + hy a + hz [ d 5] a hx a hy a + h z [ d 6] a + hx a hy a + h z [ d ] a + h a h a h 7 x y z In case ese vectors need to be represented n baseframe, ey can be calculated as[ d j ] Q[ d j ], where e matrx Q s gven by (36). he matrces and O n (37) are e 3 3 dentty and zero matrces, whereas matrx L s 3 3 matrx assocated w e angular velocty of e spatal pendulum represented as ω w e rates of e ZYZ Euler angles represented as θ ɺ ɺ φ ɺ θ ɺ ϕ,.e., ω = Lθ ɺ (38a) where e 3 3 matrx L has e followng representaton [3]: 0 Sφ SθCφ L 0 Cφ Sθ Sφ (38b) 0 Cθ B. Convectve Inerta terms he convectve nerta terms were calculated usng (35) as, C N ɶ MN ɶ ɶɺ, where, N ɺɶ DN ɺ l N d + DNɺ l N d + DN l Nɺ d n whch e elements of D ɺ were calculated as [ d ɺ j ] Qɺ [ d j ], and N ɺ l s e 6 6 matrx of zeros. he matrx N ɺ d has e followng expresson: Lɺ Nd (39) O C. Generalzed forces he generalzed forces were calculated usng (35) as τ = N ɶ f ɶ, where f ɶ ncludes externally appled forces and ose due to gravty actng at each pont-mass. he E G expresson of force s gven by fɶ = f + f n whch e E G -dmensonal vector f and f are as follows: E G f [ 0,..., 0] and [ m,..., m ] f g g (40) 7 (a) Jont angles (b) Jont rates Fg. 4.Smulaton correspondng to planar behavor Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec

6 (a) Jont angles (b) Jont rates Fg.5. Smulaton for spatal behavor where, g [ 0 0 g] = [ ] τ Nɶ f + Nɶ f τ + τ. E G G Hence, D. Smulaton Smulaton was performed usng (35) and e correspondng expressons derved n Secton V. he nput parameters n terms of e pont-masses were calculated n e begnnng of Secton V. Snce e free-fall smulaton was obtaned, no nput torque,.e., τ = 0, was consdered. he acceleraton due gravty s g=9.8 m/s. he ntal jont angles n terms of e ZYZ Euler angles were taken as φ = 90, θ = 90, ϕ = 90, whereas jont rates were taken as zeros,.e., θ ɺ (0) = 0. Fg. 4 shows e smulaton results. Snce e confguraton corresponds to a planar sold pendulum, as analyzed n [3], e results show exact match, us, apparently valdatng e algorm based on e pont-mass system. o study e spatal behavor, anoer set of ntal condtons were consdered, whch are: φ = 90, θ = 45, ϕ = 45,whereas jont rates were taken as zeros,.e., θ ɺ (0) = m/s. he resultng plots are shown n Fg. 5 ndcatng clearly spatal behavor of e pendulum. VI. CONCLUSIONS In s paper, dynamc modellng meodology for spatal multbody systems s presented usng e equmomental system of pont-masses, where each body s represented as a set of rgdly connected seven pontmasses. he correspondng pont-mass matrx and e DeNOC matrces were derved from e veloctes of e pont-masses. A numercal example s provded w a spatal pendulum where e jont moton has 3-DoF. he derved matrces were used to form e mnmal set of constraned equatons of moton from e unconstraned Newton s equatons of lnear moton. Smulaton of e spatal pendulum was performed usng MALAB s ode45 functon. ACKNOWLEDGMEN he fnancal support gven to e frst auor from e project enttled Desgn and Development of Maematcal Model based 6-DoF Electrcal Moton Platform sponsored by e Smulator Development Dvson, Secunderabad s duly acknowledged. REFERENCES [] J. J. McPhee and S. M. Redmond, Modellng multbody systems w ndrect coordnates, Computer meods n Appled Mechancs and Engneerng, vol. 95, 006, pp [] P. E. Nkravesh, Newtonan based meodologes n multbody dynamcs, n Proceedng of e nsttuton of Mechancal engneers, part K: Journal of multbody dynamcs,008, pp [3] E. J. Rou, reatse on e dynamcs of a system of rgd bodes: Elementary part-i. Dover publcaton Inc. New York, 905. [4] R. A. Wenglarz, A. A. Fogarasy and L. Maunder, Smplfed dynamc models, Engneerng 08, 969,pp [5] J. Garca de Jalon, J. Unda, A. Avello, and J. M. Jmenez, Dynamc analyss of ree-dmensonal mechansms n natural coordnates, ransacton of ASME, vol. 09, 987, pp [6] H. A. Atta, A matrx formulaton for e dynamc analyss of spatal mechansm usng pont coordnates and velocty transformaton, Acta Mechanca, vol. 65, 003, pp [7] H. A. Atta, Dynamc model of mult-rgd-body systems based on partcle dynamcs w recursve approach, Journal of Appled Mechancs, 005, pp [8] H. A. Atta, A recursve algorm for generatng e equatons of moton of spatal mechancal systems w applcaton to e fvepont suspenson, KSME Internatonal journal, vol. 8, no. 4, 004, pp [9] H. Chaudhary and S. K. Saha, An optmzaton technque for e balancng of spatal mechansms, Mechansm and Machne heory, vol. 43, 008, pp [0] H. Chaudhary and S. K. Saha, Mnmzaton of constrant forces n Industral manpulator, IEEE Internatonal Conference on Robotcs and Automaton, Rome, Italy, 007,pp [] H. Chaudhary and S. K. Saha, Dynamcs and balancng of multbody systems. Sprnger-Verlag, Berln, 009. [] S. K. Saha, Dynamcs of seral multbody systems usng e decoupled natural orogonal complement matrces, ransacton of ASME, vol. 66,999, pp [3] S. K. Saha, Introducton to robotcs. ata McGraw Hll, New Delh, 008. Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec

7 [4] V. Gupta, S. K. Saha and H. Chaudhary, Dynamcs of seral-chan multbody systems usng equmomental systems of pont-masses, Multbody Dynamcs ECCOMAS hematc Conference, Zagreb, Croata, 03, pp Proceedngs of e st Internatonal and 6 Natonal Conference on Machnes and Mechansms (NaCoMM03), II Roorkee, Inda, Dec