SPATIAL KINEMATICS OF GEARS IN ABSOLUTE COORDINATES

Transcription

1 SATIAL KINEMATICS OF GEARS IN ABSOLUTE COORDINATES Dmtry Vasenko and Roand Kasper Insttute of Mobe Systems (IMS) Otto-von-Guercke-Unversty Magdeburg D-39016, Magdeburg, Germany E-ma: KEYWORDS Dynamcs, Mutbody, Gears, Jacoban, Absoute Coordnates, Concave-convex, Rack, non ABSTRACT In ths artce the descrpton of some types of spata gear constrants (spur gears, beve gears, etc.) n absoute coordnates s consdered. Instead of the numerca expensve cacuaton of constrant Jacoban matrx we propose to use the transformed Jacoban matrx, whch can be cacuated much more effcenty. The proposed methods of descrpton of gear constrants were mpemented for the smuaton of dynamcs of CAD mode of KUKA KR 15/2 ndustra manpuator. INTRODUCTION Gears and gearng systems are fundamenta mechanca components, wdey used n the desgn of machnes and mechanca systems for the transmsson of moton and forces. On the other hand the detaed descrpton of gear constrants s usuay out of nterest of mutbody terature. In a few books can be found the descrpton of panar knematcs of gears (Haug 1989, Shabana 2001) or the descrpton of the spata gear knematcs n ont coordnates (Schweger and Otter, 2003). In ths artce the descrpton of some types of spata gear constrants (spur gears, beve gears, etc.) n absoute coordnates s consdered. The use of absoute coordnates heps to ntegrate dynamc smuaton toos wth CAD systems, wdey used for desgn of mechanca systems. In order to better understand the ssues nvoved, t s usefu to consder the equatons of moton n absoute coordnates. Let q = x T, e T T be the vector of absoute coordnates of the -th body consstng of poston coordnates x = x 1, x 2, x 3 T and of orentaton coordnates e. Orentaton coordnates can be defned n dfferent ways (e.g. Euer anges, Bryan anges, Rodrguez parameters, Euer parameters, etc.). The vector of generazed veoctes v = x T, ω T T ncudes near veocty x and anguar veocty ω. The frst dervatve of q s proportona to the vector v : q =T q v, where T denote the reaton matrx. Let q = q 1T q nt T be the vector of absoute coordnates of a mutbody system. By g(q) denote the vector of constrants, descrbng onts, connectng bodes n the smuated mechanca system. By G(q) denote the Jacoban matrx of g(q): G(q)= g(q) q (1) Dfferentatng g(q), we get the equatons of ont constrants on the veocty eve: g v (q, v)=g(q) q = G(q) T(q) v (2) Let G(q)=G(q) T(q) be the transformed Jacoban matrx. Then (2) can be wrtten as g v (q, v)=g(q) v=0 (3) We proposed (Vasenko and Kasper 2009) the method of the smuaton of mutbodes, based on the Newton-Euer equatons of moton M(q)v + G T (q)λ = f(q, v) (4) where f(q, v) s the vector of externa forces, M s the mass matrx, λ s the vector of Lagrange mutpers. The man advantage of ths method s that t uses the matrx G(q), whch can be cacuated much easy than the orgna Jacoban matrx G(q). Furthermore, f the non-mnma set of coordnates s used (e.g. Euer parameters), then the sze of G s ess than the sze of G, that s mportant for the reducton of the smuaton numerca costs. In ths artce s shown the generaton of constrants equatons g and of transformed Jacoban matrx G for some types of gear onts, commony used n mechanca systems (spur gears, beve gears, etc.). In the descrpton of gears we assume that constrants, generated by gear onts, mt ony the reatve rotaton of gears. A other mtatons on the reatve moton of connected gears (e.g. constant dstance between axes n the spur gear ont, etc.) are acheved as the resut of connecton of gears by other onts (usuay by revoute ont) to some basement. Ths art of defnton of gear constrants ooks natura and smar to the defnton of

2 gear constrants n CAD-ke systems (Autodesk Inventor, etc.). The proposed methods of descrpton of gear constrants were mpemented for the smuaton of dynamcs of CAD mode of KUKA KR 15/2 ndustra manpuator. SATIAL KINEMATICS OF GEARS Spur gear ont Equaton of constrant on the coordnate eve Let us consder the gear and the gear, shown n Fgure 1, whch ro reatve to each other about parae axes a and a. Let C and C be the ponts on axes a and a, yng on the ne, perpendcuar to a. We assume that the moton of gears s constraned n such way that ony the reatve rotaton of them s aowed,.e. a reman parae to a, the vector C C remans perpendcuar to a and the dstance between C and C reman constant (equa to the sum of the gears rad). a C p α α Fgure 1: Spur Gear Jont p By c and c denote the oca poston vectors of C and C, respectvey. Let and be the ponts of contact on bodes and, yng on the ne C C. By and denote the ponts of contact on gears at nta stage. Gears ro reatve to each other wthout sp, therefore, the arc ength and of contact on the gears must be equa. Then we get the equaton of constrant g = α r + α r (5) where α s the ange between and, α s the ange between and, r and r are the rad of gears. Now we need to fnd the formua for the cacuaton of α and α. Let, p be the vectors C and C expressed n body. Let, p be the vectors C and C expressed n body. From the concdence of ponts and foows that x + R c + R p = x + R c + R p (6) where x and x are the vector of poston coordnates of bodes and, respectvey; R and R are the transformaton matrces of the two bodes. Ths can be aso wrtten n another form C x + c + p = x + c + p (7) a c O where c = R c, c = R c, p = R p, p = R p are the vectors c, c, p, p, expressed n the goba frame. Let denote the vector of the constant ength from C to C, cacuated as Then (7) can be rewrtten as = x + c x c (8) p p = (9) Let e, e be the unts vectors aong p and p, correspondenty p = r e =, (10) From the defnton of vectors p and p foows that e and e can be cacuated as e = e = In practce the ont s usuay defned by the gear rato (11) k = r /r (12) Then r, r can be cacuated from k usng the formua r = = k r = = k 1 + k (13) (14) If the moduus of α and of α are ess than π/2, then α and α can be cacuated as α = asn a T e 0 e =, (15) where a = R a s the axs a, expressed n the goba frame. In practce we can guarantee that α and α are ess than π/2 f durng the smuaton we aways move the ponts and to the tops of ast contacted teeth. The number of teeth of gears can be defned manuay durng the gear desgn or automatcay by the smuaton pre-comper. The equaton of constrant on the veocty eve Let us show how we can fnd the equaton of constrant on the veocty eve g v and to derve from g v the formua for G. The equaton of constrant on the veocty eve are cacuated as the tme dervatve of (5) g v = α r + α r = 0 (16) Let us defne the reatve orthogona system of coordnates C e bx e by e bz where the orgn of the reatve system s rgdy connected to C, x b s the axes yng on the ne C C, e by es on the axes a (.e. e by = a ) and the vector e bz s chosen n such way that the C e bx e by e bz w be rghthanded. Let u b, u b be the veoctes of ponts and reatve C e bx e by e bz, expressed n the goba coordnate system, respectvey. From the defnton of C e bx e by e bz foows that = α a =, (17)

3 where be the reatve anguar veocty of -th body, expressed n the goba coordnate system, cacuated as = ω =, (18) where s the anguar veocty of the frame C e bx e by e bz and ω s the anguar veocty of the -th body. Cross-mutpyng both parts of (17) by the vector e, we get e = α a e =, (19) Let s be a vector perpendcuar to the axs of rotaton a and to p : s = a e. Then from (19) foows that α = s T e α = s T e (20) Substtutng α, α n (16), we get g v = s T p + s T p (21) Usng the formua (18) for,, we obtan g v = s T ω p ω p + (p p ) (22) Let u C, u C be the absoute veoctes of ponts C and C, respectvey, cacuated as u C = x + ω c =, (23) From the defnton of C e bx e by e bz foows the reaton (p p ) = u C u C (24) Substtutng ths equaton n (22) and usng (23), we get: g v = s T x + ω (c + p ) x + ω (c + p (25) ) The physca meanng of ths equaton s the equaty of proectons of veoctes of and on the axs s. Usng the trpe product formua a T (b c) = c a T b = a b T c (26) we get from (25) g v =s T x s c + p T ω s T x + s c + p T ω = 0 Now, usng (3), we get the matrx G(q,q ) s s c + p s s c + p (27) (28) Cear, that the cacuaton of G from (28) s much easer than the cacuaton of Jacoban G(q,q ) from (1). Beve gear ont Let us consder the gear and the gear, shown n Fgure 2, whch ro reatve to each other wth ntersectng axes a and a n such way that the anges of rotaton α, α are reated as α = α k (29) where k s the gear rato defned by the quotent between the number of gear teeth. Fgure 2: Beve Gear Jont In practce the ont s defned by the defnton of bodesfxed axes a and a and by the oca poston vectors c and c of some ponts C and C on the axes a and a, respectvey. By denote the vector of the constant ength from C to C, cacuated from (8). Let us defne for each body some vector p as perpendcuar to a, whch begns at C and ends at the pont, yng on the ne of contact. Let r and r be the moduus of p and p, respectvey (smary to the rad of r and r of spur gears). Let us assume that C and C are chosen n such way that ponts and are concdent,.e. p p = (30) Then from the defnton of p and p foows that It can be shown that a C k = p r = p p at r = a a r = kr 0 α (31) r Now we can reformuate the equaton of beve gear constrant (29) n the form smar to the equaton of the spur gear constrant g = α r + α r (32) Let us show now how to cacuate the anges α and α. By 0 and denote the ponts and at nta stage and by p 0, the start vaues of vectors p, p, respectvey. Obvousy, the ange of reatve rotaton α s the ange between and p and α s the ange between p 0 and p. From the numerca pont of vew t s easy to cacuate α as the ange between unt vectors e = p /r, e 0 = /r (=,). a p C α

4 From the defnton of vectors p, p foows that e = s a and e = s a, where s s a unt vector, perpendcuar to a, a, cacuated as Rack and pnon ont s = a a a a (33) If the moduus of α and of α are ess than π/2, then α and α can be cacuated from (15). Usng the same procedure for the generaton of equaton of constrant on the veocty eve, as t was used above for the spur gear case, we obtan that the transformed Jacoban matrx G can be cacuated from (28). C a C α p d a Concave-convex gear ont Fgure 3: Concave-Convex Gear Jont Let us consder a concave-convex gear ont, descrbng the rong contact of smaer gear makes nsde the arger nteror gear, shown Fgure 3. Usng the defntons from the spur gear case, we obtan the foowng reaton between anges of rotatons α r α r =0 (34) Cear, that unke convex gear case now the vectors e e p a C α are equa and are cacuated as e = e = and (35) The formua for the cacuaton of rad r and r from the gear rato k = r /r aso changes as r = = 1 1 k r = = k 1 k The anges α and α n (34) can be cacuated from (15) Usng the same procedure, as t was used above for the spur gear case, we obtan the matrx G p α a C s s c + p s s c + p (36) Fgure 4: Rack and non Jont Let us consder a crcuar gear (caed rack) and a fat bar (caed pnon) consttutng the constrant knematc par, shown n Fgure 4. The rack ros reatve to the pnon about the axs a whereby the pnon axs a s stuated on the ne of contact. It s assumed that a and a are perpendcuar n space,.e. a T a = 0. Let C be the pont of contact on pnon at nta stage and C be the ponts on axs a, chosen n such way that C C s perpendcuar to a. As before, we denote by and the ponts of contact on bodes and, and by the pont of contact on body at nta stage. The equaton of our constrant can be wrtten as g = d α r (37) where α s the ange between and, d s the dstance between and. Let us show now how to cacuate d and α. Let d be the vector from to. It s obvous that d = d a (38) Let, p, c, be the oca postons of,, C, respectvey. From the concdence of ponts and foows that x + c + p = x + c + d (39) From the defnton foows that p s perpendcuar to a. Therefore, mutpyng (39) by a, we get the formua for the cacuaton of d d = a T (x + c x c ) (40) Let e = p /r be the unt vector aong p. Let e be the vector of coordnates of e n the pnon. Obvousy, e s constant. The ange α can be cacuated from (15), whereby the vector e can be cacuated as e = R e. In the same way as n the spur gear case, we obtan G a a a p c a x + c x p (41)

5 where p can be cacuated from the equaton p = r e. SIMULATION EXAMLE: INDUSTRIAL MANIULATOR KUKA KR 15/2 In the ast years we deveoped a component-orented smuaton software Vrtua Systems Desgner (VSD), ntegrated wth CAD-ke too Autodesk Inventor (Kasper et. a 2007, Vasenko and Kasper 2007). The proposed methods of descrpton of gear ont constrants were mpemented n VSD. As a test exampe we used an Autodesk Inventor mode of the ndustra manpuator KUKA KR 15/2, shown n Fgure 5. Ths s a sx-axs robot wth artcuated knematcs for a contnuous-path controed tasks. The man areas of appcaton of KR 15/2 are handng, assemby, machnng, etc. (Specfcatons of Robots) numerca error of gear constrants on the coordnate and on the veocty eves are equa to the accuracy of used numerca methods. The anayss of montored vaues of bodes veoctes and acceeratons show the correctness of proposed methods for the generaton of gear constrants. CONCLUSION AND FUTURE WORK In ths artce s consdered the spata knematcs of most commony used types of gear onts (spur gears, beve gears, concave-convex gears and rack and pnon onts) n absoute coordnates. We show the methods of generaton of equatons of correspondent constrants on the coordnate and on the veocty eve. In standard methods of smuaton of mutbodes the cacuaton of constrant Jacoban matrx G s needed, whch n the case of gears s a numercay expensve procedure. That s why we propose to use the smuaton agorthm based on the cacuaton of transformed Jacoban matrx G. In ths artce s shown that the generaton of matrx G s easy n use and requre a very sma amount of computatona effort. Moreover, the addtona advantage G s ts reduced sze n comparson wth G when the non-mnma set of orentaton coordnates s used. The proposed methods of descrpton of gear constrants were mpemented for the smuaton of dynamcs of KUKA KR 15/2 ndustra manpuator. Test resuts show the correctness of proposed agorthms. In future we pan to mprove the area of mpementaton of proposed method by the descrpton of other moton transmton eements (e.g. bet onts, cam-foowers, etc.). Fgure 5: CAD Mode of KUKA KR 15/2 The compete Autodesk Inventor mode ncudes 1036 parts. The correspondent VSD mode conssts of 43 bodes connected by 95 onts (ncudng 7 spur gear onts, 3 beve gear onts and 10 concave-convex gear onts). Some of mode constrants are redundant because of the mode s desgn n Autesk Inventor (e.g. the defnton of stff connecton as three pane-to-pane onts eads to the generaton of three redundant constrants). The dynamcs of the manpuator under the acton of gravtatona force and of torques n motors s smuated. The REFERENCES Haug, E Computer Aded Knematcs and Dynamcs of Mechanca Systems Voume I: Basc Methods. Ayn & Bacon, Boston. Kasper, R.; D. Vasenko and G. Sntotsky A Component Orented Approach to Mutdscpnary Smuaton of Mechatronc Systems. roceedngs of the EUROSIM Congress on Modeng and Smuaton (EUROSIM 2007), September 9-13 Lubana, Sovena. Specfcaton of Robots KR 6/2, KR 15/2, KR 15 L6/2, by KUKA Robot Group. Schweger, C. and Otter, M Modeng 3D Mechanca Effects of 1D owertrans, n roceedngs of the 3rd Internatona Modeca Conference,. Frtzson, ed., Lnkopng, November 2003, The Modeca Assocaton and Lnkopng Unversty, pp Shabana, A.A Computatona Dynamcs, Wey, New York, Vasenko, D. and R. Kasper Integraton Method of CAD Systems roceedngs of the ASME 2007 Internatona Desgn Engneerng Technca Conferences & Computers and Informaton n Engneerng Conference IDETC/CIE 2007 September 4-7, 2007, Las Vegas, Nevada, USA Vasenko, D. and R. Kasper Successve proecton method for the smuaton of spata dynamcs of mutbodes. roceedngs of Mutbody Dynamcs 2009 (ECCOMAS Thematc Conference), Warsaw, oand, June 29-2 Juy, 2009 (accepted, to appear)