Intelligent Dynamic Simulation of Mechanisms

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Intellgent Dynamc Smulaton of Mechansms Glenn A. Kramer Schlumberger Laboratory for Computer Scence 8311 North RR 620 Austn, Texas 78726 net: gak~slcs.slb.

1 Intellgent Dynamc Smulaton of Mechansms Glenn A. Kramer Schlumberger Laboratory for Computer Scence 8311 North RR 620 Austn, Texas net: gak~slcs.slb.com Abstract Dynamc smulaton nvolves solvng sets of equatons that represent knematc (or geometrc) and knetc (or force related) constrants. The knematc equatons are hghly nonlnear algebrac equatons, and the knetc equatons are coupled dfferental equatons that must be ntegrated over tme. In most smulators, the algebrac and dfferental equatons are solved smultaneously n an teratve manner usng sparse matrx techques and stff ntegraton schemes. Ths s not partcularly effcent, and can also lead to numercal stablty problems. I propose an alternatve formulaton, based drectly on the work of Euler and Lagrange, whch allows more effcent soluton of dynamcs problems at nteractve rates. Ths paper descrbes the use of a geometrc constrant engne based on degrees of freedom analyss (developed for the knematc smulator TLA), wth extensons to ncorporate reasonng about dynamcs. I present a method for calculatng velocty ratos, formulated as a problem n geometry. These calculatons, n conjuncton wth the knematc soluton, allow determnaton of all knetc and potental energy terms requred for dynamc smulaton. The constructed system of dynamcs equatons are pure dfferental equatons n terms of a mnmal set of generalzed coordnates. The formulaton results n small, dense matrces, rather than large, sparse ones. In prncple, large tme-steps may be used for coarse dynamc behavor, wth smaller tme-steps yeldng better approxmatons to the true behavor, at the cost of more computaton tme. Introducton Understandng complex mechancal devces requres the ablty to smulate the behavors of the devces, and to make reasonable generalzatons about those behavors. Mechansm smulaton may be done at a purely qualtatve level [Km, 1990], a mxed numercal/quatatve level [Joskowcz and Sacks, 1991], or at a detaled numercal level followed by qualtatve characterzaton of the results [Gelsey, 1990]. Each of these approaches has advantages and dsadvantages wth regard to predctve power, generaton of ncomplete or mpossble behavors, and computatonal cost. Detaled numercal smulaton provdes the most accurate level of detal, but at the hghest cost. Snce multple smulatons are needed to extract data about trends (e.g., velocty vs. change n a parameter value), precse numercal smulaton s often unattractve, partcularly at the earlest stages of desgn. In ths paper I propose an ntellgent computatonal methodology for performng accurate dynamc analyss of rgd-body mechansms n an effcent, nteractve manner. Ths methodology wll allow fast evaluaton of desgn alternatves, and provde nformaton for senstvty analyss and force analyss. The approach separates the problem nto knematcs and knetcs, as frst proposed by Euler [Hartenberg and Denavt, 1964]. The knematc analyss s effcently computed usng degrees of freedom analyss [Kramer, 1992]. Velocty analyss, requred for knetcs, s performed usng screw theory [Ball, 1900], formulated as another geometrc constrant problem. Ths analyss allows easy dervaton of the Lagrangan form of the dynamcs equatons n terms of a mnmal set of generalzed coordnates, thereby resultng n effcent and stable computaton methods. In addton, by adjustng the tme-step, coarse graned dynamc analyss s possble n an even more effcent manner. Related work Qualtatve knematcs and dynamcs Smple processes and mechancal systems have been descrbed n a number of qualtatve reasonng systems [Weld and de Kleer, 1990]. Most systems handle only the smplest cases of nonlnearty, due to the coarse structure of the qualtatve representatons. More specalzed qualtatve reasonng, lke trgonometrc reasonng, can gve more precse results, such as whether or not a partcular lnk n a mechansm can rotate completely wth respect to another [Km, 1990]. It cannot be used to descrbe specfc attrbutes of the space curves traced by arbtrary ponts on the mechansm, 62

2 yet ths nformaton s essental to desgners. As the descrpton of the model s more fnely dscretzed, more detal can be obtaned from the smulaton. Confguraton spaces have been used to model a varety of mechansms [Joskowcz and Sacks, 1991]. The representaton becomes computatonally ntractable for more complcated non-fxed-axs mechansms. However, wthn ts doman of applcablty, the confguraton space approach s able to deal wth topology changes durng the operaton of the mechansm. Smple approxmatons to dynamc behavor, ncorporatng models of steady-state forces, allow a number of mechansms to be smulated at a coarse dynamc level [Sacks and Joskowcz, 1992]. Detaled numercal smulaton, followed by abstractng the results nto qualtatvely nterestng regons, s the most accurate and general approach to descrbng dynamcs n qualtatve terms [Gelsey, 1990]. However, the process can be tme-consumng due to long runtmes for general-purpose smulators. Snce many smulaton runs may be needed to extract data about behavoral trends, ths approach can be computatonally nfeasble. Matrx-based dynamcs methods In the matrx-based approach to dynamcs, the knematc equatons and the dfferental equatons are solved together. One approach s to use a maxmally redundant set of generalzed coordnates (sx postonal and sx velocty per rgd body), and solve usng sparse matrx technques [Orlandea et al., 1977]. The resultng equatons are stff, and therefore requre small ntegraton tme-steps. Other approaches attempt to reduce the number of generalzed coordnates n order to mprove computatonal effcency, and to help reduce the stffness of the matrx [Hang, 1985]. General-purpose smulators usually deal wth fxed topology mechansms. Cremer descrbes a smulator wth bult-n capabltes for detectng collsons and other topologcal changes. Hs smulator reformulates the equatons when changes occur, and then contnues dynamc smulaton [Cremer, 1989]. Symbolc dervaton of dynamcs equatons An alternatve to detaled numercal soluton of the mxed algebrac/dfferental equatons s the symbolc generaton of a set of pure dfferental equatons, whch descrbe the system n terms of a mnmal set of generalzed coordnates correspondng to the system s true degrees of freedom. The Dyne system [Brown and Lefer, 1991] uses symbolc reasonng to derve such equatons, guded by a set of algebrac transformaton rules and meta-level control rules. Such systems are hard to desgn, mantan, and debug. However, the equatons derved usng Dyne are useful to desgners performng senstvty analyss at selected ponts n the mechansm s behavor, and for descrbng qualtatve regons of behavor. Symbolc geometrc soluton of knematcs problems usng degrees of freedom analyss s descrbed n [Kramer, 1992], wth extensons to other knds of geometry presented n [Kramer, to appear]. The next secton extends ths technque to generate dynamcs equatons more effcently than rule-based systems whch drectly manpulate algebrac and dfferental equatons. A drect method for dynamcs Drect methods for dynamcs can be traced to Euler, who advocated treatng dynamcs by parttonng the problem nto two parts: knematcs and knetcs [Hartenberg and Denavt, 1964]. Knematcs deals wth the postons of the parts of the mechansm as constraned by geometrc relatonshps. Relatve (but not absolute) veloctes and acceleratons often can be calculated knematcally. Knetcs deals wth how physcal objects move under the effect of forces, and deals wth absolute velocty, acceleraton, mass, nerta, etc. Euler also demonstrated how an object n threedmensonal space can be moved from one poston to an arbtrary second poston by a combnaton of a sngle translaton and a sngle rotaton, where the rotaton axs s parallel to the translaton vector, resultng n a screw-lke moton. At any nstant n tme, a body n moton may be thought of as movng about an nstantaneous screw n space; ths screw s poston, orentaton, and ptch changes over tme. The theory of screws was treated n depth n a geometrc manner n [Ball, 1900]. Wth the nstantaneous screws known for each body n a rgd-body mechansm, all relatve veloctes n the system are related by ratos of dstances from the approprate screws. Specfyng one absolute velocty then allows fndng all absolute veloctes. The veloctes provde the knetc coenergy terms n the lagrangan formulaton of dynamcs, and the knematc nformaton provdes the potental energy terms [Crandall et a/., 1982]. All terms requred n the lagrangan are attanable from smple geometrc constructons. Ths leads to the followng algorthm for constructng the lagrangan and usng t n dynamc analyss: 1. Calculate the knematc nformaton n terms of the generalsed poston coordnates. 2. Fnd the nstantaneous screw axes by geometrc constructon. 3. Calculate angular and lnear veloctes as ratos of dstances betwen ponts on the mechansm and the screw axes. 4. Calculate the tme dervatves of the screw axes. 5. Use the above nformaton to construct the lagrangan drectly. 6. Solve the lagrangan for acceleratons, and ntegrate over tme.

3 J2 12,,am z J3,,, r1,,",r 3 /v ",, S... o.. -XII... Fgure 1: A four-bar lnkage. 0) 1 The remander of ths paper descrbes the soluton of a smple dynamcs problem usng ths algorthm. At present, the mathematcs have been solved, but the soluton process has not been automated. A four-bar lnkage Fgure 1 descrbes a four-bar planar lnkage. One of the lnks s grounded, or fxed to the global reference frame, and s not shown. The movng lnks have lengths 11,12, and/3, and are modeled wth masses ml, m2, and m3 and rotatonal nertas I1,I~, and In. The revolute jonts jl and j4 connect the cranks to the ground; the remanng two jonts connect the cranks to the coupler. The lnkage has one degree of freedom; the poston of all lnks are determned fully by the crank angle 0, and the veloctes by the frst dervatve of 0 wth respect to tme (0). The example problem s an ntal values problem. Gven ntal values for 0 and 0, fnd the behavor of the lnkage over tme. In ths problem, gravty exerts a downward force, as llustrated, and the jonts are assumed to be frctonless. Computng angular and lnear veloctes The frst portons of the algorthm requre calculatng the knematc nformaton as a functon of the generalzed poston coordnates, fndng the nstantaneous screw axes by geometrc constructon, and calculatng angular and lnear veloctes as ratos of dstances of mass centers from the screw axes. The frst tem s covered by knematc analyss [Kramer, 1992]; the next two are covered here. In planar mechansms, the axes of all nstantaneous screws are normal to the plane, and the ptch s always zero. In ths specalzed case, the nstantaneous screw s known as the nstantaneous center of moton [Hall, 1961]. Fgure 2 llustrates how the nstantaneous centers and veloctes are calculated. The nstantaneous centers of the cranks are the fxed jonts (jl for lnk 1, and j4 for lnk 3); these centers do not change over tme. ~,? Jl J4 ~3 Fgure 2: Knetc analyss of the lnkage. The nstantaneous center for lnk 2 s computed geometrcally as follows. Snce jont j~ s on the lnk, t must rotate about the nstantaneous center for lnk 2. However, snce j2 s also on lnk 1, t must rotate about jl, and hence, ts lnear velocty must be perpendcular to the lne descrbng lnk 1. Therefore, the nstantaneous center must le on a lne through jl and j2. Usng a smlar argument for the moton of jont j3, the nstantaneous center s found by ntersectng the two lnes. Lnk 2 rotates about the ntersecton pont wth angular velocty w2, whch s yet to be determned. The angular velocty of lnk 1, wl, s specfed to be 0. The angular velocty 0~2 s found by equatng the lnear veloctes at jont j2: ~all = -taarl. In smlar fashon, the remanng angular veloctes are calculated: wl = 0 (1) w~ = -~1(l/rl) = - 01(11/rl) (2) w3 = -w2(ra/h) = 01(llra/larl) (3) The drectons of the lnear veloctes are derved from the angular veloctes. The magntudes of the lnear veloctes (assumng each center of mass s at the center of the lnk) follow drectly from the angular veloctes: I,,11= w1(/1/2) -- 0"1(/1/2) (4) 1"21= co~r~ = 01(llr21rl) (5) Iv31--,.3(h/2) = 01(llra/2rl) (6) Thus, all angular and lnear veloctes are calculated as smple ratos of dstances between nstantaneous centers and ponts on the mechansm. For a more compact notaton, the ratos are wrtten as w = 0k, and Ivll = Oj. 64

4 Generatng the lagrangan The lagrangan of the system s then descrbed as follows (assumng a postve y axs movng upward n Fgure 2, and wth v = [vd: 1 Here, h s the moment of nerta for mass lnk, and g s the acceleraton of gravty. The dfferental equaton descrbng the moton of the lnkage s then [Crandall et al, 1982]: "r l o,: d~ LaOJ - "~" = 0 (8) The second term n Equaton 8 s the potental energy term, and s calculated as follows: 0_.~ 0y, 00 = -g = (9) The change n heght (y) for each pont mass s: Oy~ Op~ o0 = o-t ~ (10) where Op/O0 s the change n poston of pont mass rn~ wth respect to a O. Ths vector s always n the drecton of v. The value of Opz/OO s calculated from O as follows: 0~1 =_~ (~)sno+~r (~)coso The remanng poston change magntudes are related to each other drectly as the magntude of the lnear veloctes, whch have already been computed geometrcally: I~1 Ivd 10pl- I*l (12) The frst term n Equaton 8 s the knetc coenergy term, and s calculated as follows. The dervatve wth respect to 0 s: 0 x-~,! Ov The tme dervatve of ths quantty s: O, o (13) (14) of an nstanta- Fgure 3: Calculatng the dervatve neous center. The frst term nvolves the lengths of geometrc enttes already constructed. Evaluatng the second term nvolves understandng how the nstantaneous centers move over tme. In ths example, the only nstantaneous center that moves over tme s the center for lnk 2. Tme dervatve of the nstantaneous center Usng the chan rule, fndng the tme dervatves of the j s (as well as the k s) s reduced to fndng the dervatves of these quanttes wth respect to 0: dj ded0 "d~. dt -- "~"dt = 0arp (15) The quantty djl/do may be found through geometrc senstvty analyss, nvolvng smple geometrc constructons, x Fgure 3 llustrates the calculaton. Consder three ponts Pl,P~, and P3, where ponts Pl and P3 are fxed n space. If the lne through PlP~ s rotated through a small dsplacement 501, and p2 s constraned to le on psp2, t wll move n drecton 5p~,x. If, on the other hand, lne P3P~ rotated by a small dsplacement 683, P2 wll move n drecton 5p2,s. For a composte change of both 50z and 603, the movement of pont p~ wll be the vector sum of the two ndependent movements (for nfntesmal dsplacements, the quadrangle on whch the three vectors le becomes a parallelogram). In the case where PI and P3 are the grounded jonts of the four-bar lnkage, and pont pu s the nstantaneous center of lnk 2, the length PP2 s!~ + r~, and P3Pu s/3 + ra. Snce 1 and h are parameters of the lnkage, and constant, the value of 5pu yelds 5rz and 5ra trvally. The only remanng detal s to determne 603 n terms of 501, so only one ndependent quantty ~Work on geometrc senstvtes s beng pursued by Jahr Pabon at Scldumberger. Ths secton employs hs technques. 65

s used when takng the dervatve. The rato of the and [Sacks and Joskowcz, 1992] wll be explored as angular veloctes provdes ths nformaton: possble means of dealng wth changeable topology.

5 s used when takng the dervatve. The rato of the and [Sacks and Joskowcz, 1992] wll be explored as angular veloctes provdes ths nformaton: possble means of dealng wth changeable topology. Besdes ther use n formulatng the dynamcs equatons, 501 = ~.~.I 60s ~s (16) the geometrc senstvtes can also be used for force analyss at selected ponts n the mechansm s Integraton of the dfferental equatons trajectory, and for knematc analyss of velocty ratos. These are quanttes that could be optmzed n a desgn at nteractve rates. If the dynamc behavors can be smulated effcently enough, t may be possble to make mult-dmensonal "maps" of smulated behavor as a functon of desgn parameter values. Such behavoral maps could be of substantal beneft n the desgn and debuggng of complex mechancal devces. Integraton schemes have yet to be explored, but snce the system of equatons s purely dfferental, wth no algebrac equatons, stffness should not be an overrdng concern. Tradtonal ntegraton schemes, such as adaptve step sze Runge-Kutta and predctorcorrector methods, wll be explored. If ntegraton uses large tme-steps, the effect should be to have "approxmate" dynamcs. There s no danger of the mechansm "flyng apart," snce the knematc constrants are not consdered n the tme ntegraton. Thus, there s lkely to be a convenent tradeoff between computaton tme, accuracy, and nteractveness. Theoretcal analyss Generatng the plan of geometrc constructons to fnd veloctes need only be done once for a gven mechansm. After that, the plan may be reused durng each step of the dynamc smulaton. Snce there are as many nstantaneous screws as there are bodes n the mechansm, evaluaton of the knetc coenergy terms of the lagrangan takes tme lnearly proportonal to the sze of the mechansm. Knematc analyss, necessary for the potental energy terms, s O(n log n), but typcally lnear n n, where n s the number of bodes n the mechansm [Kramer, 1992]. Solvng the lagrangan for the acceleratons requres nvertng a matrx of sze d, the number of true degrees of freedom n the mechansm. Ths contrasts wth the standard matrx-based approaches, where the matrx to be nverted s of sze proportonal to the number of bodes n the mechansm. In the worst case of a mechansm comprsed exclusvely of open chans, the number of true degrees of freedom wll be proportonal to the number of bodes; however, the absolute number of generalzed coordnates beng consdered wll stll be less usng the geometrc algorthm, snce the knematc constrants are already elmnated. Dscusson Ths work s n ts early stages, and much of the detal remans to be worked out. However, the geometrc constructons requred for the velocty analyss can all be performed n a hghly stylzed fashon. If the gans n computatonal effcency are comparable wth the gans n knematc smulaton n [Kramer, 1992], the speedup n dynamc smulaton could be substantal, affordng nteractve smulaton speeds for many complex mechansms. At present, only mechansms wth fxed topology are beng explored. Technques found n [Cremer, 1989] Acknowledgments I would lke to thank Jahr Pabon and George Celnker for ther contrbutons to these deas. References [Ball, 1900] R. S. Ball. A Treatse on the Theory of Screws. Cambrdge Unversty Press, Cambrdge, UK, [Brown and Lefer, 1991] D. R. Brown and L. J. Lefer. The role of meta-leve! nference n problem-solvng strategy for a knowledge-based dynamcs analyss ad. Journal of Mechancal Desgn, Trans. ASME, 113: , December [Crandall et a., 1982] Stephen H. Crandall, Dean C. Karnopp, Edward F. Kurtz, Jr., and Davd C. Prdmore-Brown. Dynamcs of Mechancal and Eectromechanca Systems. Robert E. Kreger Publshng, Inc., Malabar, Florda, Orgnally publshed: McGraw-Hll, New York, [Cremer, 1989] James F. Cremer. An Archtecture.for General Purpose Physcal System Smulaton- Integratng Geometry, Dynamcs, and Control. PhD thess, Cornell Unversty, Ithaca, New York, Aprl Department of Computer Scence TR [Gelsey, 1990] Andrew Gelsey. Automated Reasonng about Machnes. PhD thess, Yale Unversty, Aprl YALEU/CSD/RR No [Hall, 1961] A. S. Hall, Jr. Knematcs and Lnkage Desgn. Bat Publshers, West Laffayette, Indana, Republshed by Wavelength Press, Prospect Heghts, Illnos. (1986). [Hartenberg and Denavt, 1964] R. S. Hartenberg and J. Denavt. Knematc Synthess of Lnkages. McGraw-Hll, New York, [Hang, 1985] Edward Hang. Computer Aded Knematcs and Dynamcs of Mechancal Systems, volume 1 Basc Method. Department of Mechancal Engneerng, Unversty of Iowa, Iowa Cty, Iowa,

6 [Joskowcz and Sacks, 1991] L. Joskowcz and E. P. Sacks. Computatonal knematcs. Artfcal Intellgence, 51: , [Km, 1990] Hyun-Kyung Km. Qualtatve knematcs of lnkages. Techncal Report UIUCDS-R , Unversty of Illnos, Champagn-Urbana, May [Krarner, 1992] Glenn A. Kramer. Solvng Geometrc Constrant Systems: A case study n knematcs. MIT Press, Cambrdge, Massachusetts, [Kramer, to appear] Glenn A. Kramer. A geometrc constrant engne. Artfcal Intellgence, (to appear). [Orlandea et al., 1977] N. Orlandea, M. A. Chase, and D. A. Calahan. A sparsty orented approach to the dynamc analyss and desgn of mechancal systems - parts I and II. Journal of Engneerng for Industry, Trans. ASME Ser. B, 99: , , [Sacks and Joskowcz, 1992] Elsha Sacks and Leo Joskowcz. Mechansm smulaton wth confguraton spaces and smple dynamcs. Department of Computer Scence Techncal Report CS-TR , Prncton Unversty, Prnceton, N J, March [Weld and de Kleer, 1990] D. S. Weld and J. de Kleer, edtors. Readngs n Qualtatve Reasonng about Pl~ysca Systems. Morgan Kaufmann, San Mateo, Calforna, 1990.

Kinematics of pantograph masts

Kinematics of pantograph masts Abstract Spacecraft Mechansms Group, ISRO Satellte Centre, Arport Road, Bangalore 560 07, Emal:bpn@sac.ernet.n Flght Dynamcs Dvson, ISRO Satellte Centre, Arport Road, Bangalore 560 07 Emal:pandyan@sac.ernet.n