Nonlinear Effects in Stiffness Modeling of Robotic Manipulators

Transcription

1 A. Pashkevch, A. Klmchk D. Chablatonlnear effect n stffness modelng of robotc manpulators, In: Proceedngs of Internatonal Conference on Computer and Automaton echnology (ICCA 29), Vence, Italy, 29, World Academy of Scence, Engneerng and echnology 58 (29) Nonlnear Effects n Stffness Modelng of Robotc Manpulators A. Pashkevch, A. Klmchk and D. Chablat Abstract he paper focuses on the enhanced stffness modelng of robotc manpulators by takng nto account nfluence of the external force/torue actng upon the end pont. It mplements the vrtual jont technue that descrbes the complance of manpulator elements by a set of localzed sx-dmensonal sprngs separated by rgd lnks and perfect jonts. In contrast to the conventonal formulaton, whch s vald for the unloaded mode and small dsplacements, the proposed approach mplctly assumes that the loadng leads to the non-neglgble changes of the manpulator posture and correspondng amendment of the Jacoban. he developed numercal technue allows computng the statc eulbrum and relevant force/torue reacton of the manpulator for any gven dsplacement of the end-effector. hs enables desgner detectng essentally nonlnear effects n elastc behavor of manpulator, smlar to the bucklng of beam elements. It s also proposed the lnearzaton procedure that s based on the nverson of the dedcated matrx composed of the stffness parameters of the vrtual sprngs and the Jacobans/Hessans of the actve and passve jonts. he developed technue s llustrated by an applcaton example that deals wth the stffness analyss of a parallel manpulator of the Orthoglde famly. Keywords Robotc manpulators, Stffness model, Loaded modeonlnear effects, Bucklng, Orthoglde manpulator I. INRODUCION Current trends n mechancal desgn of robotc manpulators are targeted at essental reducton of movng masses, n order to acheve hgh dynamc performances wth relatvely small actuators and low energy consumpton. hs motvates usng advanced knematcal archtectures (Orthoglde, Isoglde, Delta, etc.) and lght-weght materals, as well as mnmzaton of cross-sectons of all crtcal elements. he prmary constrant for such mnmzaton s the mechancal stffness of the manpulator, whch s drectly related wth the robot accuracy defned by the desgn specfcatons. In robotc lterature, the manpulator stffness s usually A. Pashkevch s wth the Department Automatue et Productue, Ecole des Mnes de Nantes, 4 rue Alfred-Kastlerantes 4437 rance and Insttut de Recherches en Communcatons et Cybernétues de Nantes, rue de la No, 4432 Nantes, rance, (phone: fax: e- mal: Anatol.Pashkevch@emn.fr). A. Klmchk s wth the Department Automatue et Productue, Ecole des Mnes de Nantes, 4 rue Alfred-Kastlerantes 4437 rance and Insttut de Recherches en Communcatons et Cybernétues de Nantes, rue de la No, 4432 Nantes, rance, (e-mal: Alexandr.Klmchk@emn.fr). D. Chablat s wth Insttut de Recherches en Communcatons et Cybernétues de Nantes, rue de la No, 4432 Nantes, rance, (e-mal: Damen.Chablat@rccyn.ec-nantes.fr). evaluated by a lnear model, whch defnes the statc response to the external force/torue, assumng that the complant deflectons are small and the statc preloadng s nsgnfcant. However, n many practcal applcatons (such as mllng, for nstance), the preloadng s essental and conventonal stffness modelng technues must be used wth great cauton. Moreover, for the manpulators wth lght-weght lnks, there s a potental danger of bucklng phenomena that s known from general theory of elastc stablty []. Hence, the exstng stffness modelng technues for hgh-performance robotc manpulators must be revsed, n order to add ablty of detectng non-lnear effects and avod structural falures caused by the preloadng. he exstng approaches for the manpulator stffness modelng may be roughly dvded nto three man groups: the nte Element Analyss (EA) [2], the matrx structural analyss (SMA) [3], and the vrtual jont method (VJM) that s often called the lumped modelng [4]. he most accurate of them s the nte Element Analyss, whch allows modelng lnks and jonts wth ts true dmenson and shape. However t s usually appled at the fnal desgn stage because of the hgh computatonal expenses reured for the repeated remeshng of the complcated 3D structure over the whole workspace. he SMA also ncorporates the man deas of the EA, but operates wth rather large elements 3D flexble beams that are presented n the manpulator structure. hs leads obvously to the reducton of the computatonal expenses, but does not provde clear physcal relatons reured for the parametrc stffness analyss. And fnally, the VJM method s based on the expanson of the tradtonal rgd model by addng the vrtual jonts (localzed sprngs), whch descrbe the elastc deformatons of the lnks, jonts and actuators. he VJM technue s wdely used at the pre-desgn stage and wll be extended n ths paper for the case of the preloaded manpulators. It should be noted, that there are a number of varatons and smplfcatons of the VJM, whch dffer n modelng assumptons and numercal technues. Recent modfcaton of ths method allows to extend t to the over-constraned manpulator and to apply t at any workspace pont, ncludng the sngular ones [5] []. Besdes, to take nto account real shape of the manpulator components, the stffness parameters may be evaluated usng the EA modelng. he latter provded the EA-accuracy throughout the whole workspace wthout exhaustve remeshng reured for the classcal EA.

2 At present, there s very lmted number of publcaton that drectly addressed the problem of the stffness modelng for preloaded manpulators. he most essental results were obtaned n [7], [8] where the stffness matrx was computed takng nto account the change n the manpulator confguraton due to the preloadng. However, the problem of fndng the correspondng loaded eulbrum was omtted, so the Jacoban and Hessan were computed n a tradtonal way,.e. for the neghborhood of the unloaded eulbrum. he latter yelded essental computatonal smplfcaton but also mposed essental lmtatons, not allowng detectng the bucklng and other non-lner effects. hs paper presents a complete soluton of the consdered problem, takng nto account nfluence of the external force/torue on the manpulator confguraton as well as on ts Jacoban and Hessan. It mplements the vrtual jont technue that descrbes the complance of the manpulator elements by a set of localzed sx-dmensonal sprngs separated by rgd lnks and perfect jonts. he remander of the paper s organzed as follows. Secton 2 defnes the research problem and presents the knetostatc model of the manpulator. In Secton 3, t s proposed a numercal algorthm for computng of the loaded statc eulbrum. Secton 4 focuses on the stffness matrx evaluaton. And fnally, Secton 5 contans a numercal example that llustrates the nonlnear effects n the stffness behavor of manpulators. II. PROBLEM SAEMEN Let us consder a parallel manpulator whch conssts of a fxed base, several dentcal knematc chans and a moble platform. ypcal examples of such knematcs (g. ) are the 3-PUU translatonal parallel knematc machne [9], Delta parallel robot [], Orthoglde parallel manpulator [] and others [2] [3]. (a) (b) g. Knematcs of typcal parallel manpulators: (a) Orthoglde manpulator, (b) Par 2 atronk manpulator o evaluate the manpulator stffness, let us apply the VJM method that assumes that the tradtonal rgd model s expended by addng vrtual jonts, whch descrbe stffness of the actuator and lnks. hus, each chan of the manpulator can be descrbed by a seuence of the followng typcal elements: (a) a rgd lnk between the manpulator base and actuatng jont descrbed by the constant homogenous transformaton matrx Base (b) the -d.o.f. actuatng jonts defnng three translatonal and three rotatonal actuator coordnates, whch are descrbed by the homogenous matrx functon Ac ( a, a ) where a s the actve jont coordnates and a are the vrtual sprngs coordnate of the actuator (c) the -d.o.f. manpulator chan defnng by the rgd lnks, vrtual and passve jonts coordnates, whch descrbes by the homogenous matrx functon Chan ( c, c ) where c and c are the vectors, whch collects all passve and vrtual jonts coordnates of the chan respectvely (d) a rgd lnk from the last lnk to the end-effector, descrbed by the homogenous matrx transformaton ool. Hence, the end-effector poston may be computed by seuental multplcaton of the above homogenous matrces, so the knematc model of a separate chan may wrtten as (, ) (, ) () Base Ac a a Chan c c ool hs expresson ncludes both tradtonal geometrc varables (passve and actve jont coordnates) and stffness varables (vrtual jont coordnates). Explct poston and orentaton of the end-effector can by extracted from the matrx [4], so fnally the knematc model can be rewrtten as t g(, ) (2) where g s the geometry functon whch depends of the passve ( ) and vrtual jont ( ) coordnates, the vectors (, 2,..., ) n ncludes all passve jont coordnates, the vector (, 2,..., ) m collects all vrtual jont coordnates, n s the number of passve jons, m s the number of vrtual jonts. o evaluate the manpulator ablty to respond to external forces and torues, t s necessary to ntroduce addtonal euatons that defne the vrtual jont reactons to the correspondng sprng deformatons. or analytcal convenence, correspondng expressons may be collected n a sngle matrx euaton τ K (3) where ( τ,, τ,2,..., τ, m ) vrtual jont reactons, dag (,,,2,...,,m ) τ s the aggregated vector of the K K K K s the aggregated sprng stffness matrx of the sze m m, and, K s the sprng stffness matrx of the correspondng lnk. Smlarly, one can defne the aggregated vector of the passve

3 jont reactons ( τ,, τ,2,..., τn, ) τ but, n ths case, all ts components must be eual to zero τ. In general case, the desred stffness model s defned by a non-lner relaton Δt f ( ) (4) that descrbes resstance of a mechansm to deformatons Δt caused by an external force/torue []. It should be noted that the mappng Δt s strctly mathematcally defned and physcally tractable n all cases, ncludng underconstraned knematcs and sngular confguratons of the manpulator. However, the converse s not true. In engneerng practce, functon f (...) s usually lnearzed n the neghborhood of the statc eulbrum (, ) correspondng to the end-effector poston t and external loadng. or the unloaded mode,.e. when and the stffness model s expressed by a smple relaton K ( Δt, ) (5) where K s stffness matrx and the vector (, 2,..., ) n defnes the eulbrum confguraton correspondng to the end-effector locaton t, n accordance wth the manpulator geometry. However, for the loaded mode, stffness model have to be defned n the neghborhood of the statc eulbrum that corresponds to another manpulator confguraton (, ), whch s caused by external forces. In ths case, the stffness model descrbes the relaton between the ncrements of the force δ and the poston δt δ K (, t ) δ () where Δ + and + Δ denote the new poston of the manpulator, Δ and Δ are the devatons of the passve jont and vrtual sprng coordnates. Hence, the problem of the stffness modelng n the loaded mode may be dvded nto two seuental subtasks: () fndng the statc eulbrum for the loaded confguraton and () lnearzaton of relevant force/poston relatons n the neghborhood of ths eulbrum. Let us consder these two sub-problems conseuently. III. SAIC EQUILIBRIUM OR HE LOADED MODE Let us assume that, due to the external force, the endeffector of the manpulator s relocated from the ntal (unloaded) poston t P(, ) to a new poston t P(, ), whch satsfes the condton of the mechancal eulbrum. Here s computed va the nverse knematc and s eual to zero (snce there are no external loadng n the sprngs),, are passve and vrtual jont coordnate n the loaded mode respectvely. or rather small dsplacement Δt t t, a new poston of the end-effector t Δ P( + Δ, + ) may be expressed as t Δ t + J Δ + (7) where J and J are the knematc Jacobans wth respect to the coordnates,, whch may be computed from () analytcally or sem-analytcally, usng the factorzaton technue []. However, n general case, the stffness model s hghly non-lnear and computng (, ) reures some addtonal efforts. or computatonal reasons, let us consder the dual problem that deals wth determnng the external force and the manpulator confguraton (, ) that correspond to the output poston t. Let us assume that the jonts gve small, arbtrary vrtual dsplacements Δ n the eulbrum neghborhood. Accordng to the prncple of vrtual dsplacements, the vrtual work of the external force appled to the end-effector along the correspondng dsplacement Δt J Δ + J Δ s eual to the sum ( ) ( ) JΔ + J Δ. Snce the passve jonts do not produce the force/torue reactons, the vrtual work ncludes only one component τ Δ (the mnus sgn takes nto account the force-dsplacement drectons for the vrtual sprng). In the statc eulbrum, the total vrtual work of all forces s eual to zero for any vrtual dsplacement, therefore the eulbrum condtons may be wrtten as J τ J akng nto account (3), the latter can be rewrtten as J K J It s evdent that there s no general method for analytcal soluton of ths system and t s reured to apply numercal technues. o derve the numercal algorthm, let us lnearze the knematc euaton n the neghborhood of the current poston (, ) t ( J, ) + (, ) ( J ) + (, ) ( ) () P + + and rewrte the statc eulbrum euatons as (8) (9)

4 J (, ) K + + J (, ) + () hs leads to a lnear algebrac system of euatons wth respect to ( +, +, + ) J(, ) + J+ (, ) + t f (, J ) + (, ) J + (, ) + (, ) + K J+ J (, ) + whch gves the followng teratve scheme + J (, ) (, ) (, ) K J J + J (, ) t f (, J) + (, ) J + (, ) K J (, ) + + (2) (3) where the startng pont (, ) can be chosen usng the nonloaded confguraton, and computed va the nverse knematcs. As follows from computatonal experments, for typcal values of deformatons the proposed teratve algorthm possesses rather good convergence (3-5 teratons are usually enough). However, n the case of bucklng or n the area of multple eulbrums, the problem of convergence becomes rather crtcal and hghly depends on the ntal guess. o overcome ths problem, the value of the jont varables (, ) computed at each teratons were dsturbed by addng small random nose. urther enhancement of ths algorthm may be based on the full-scale Newton-Raphson technue (.e. lnearzaton of the statc eulbrum euatons n addton to the knematc one), ths obvously ncreases computatonal expenses but potentally mproves convergence. IV. SINESS MODEL OR HE LOADED MODE After the statc eulbrum correspondng to the external loadng s found, the force-dsplacement relatons may be lnearzed. o compute the desred stffness matrx, let us assume that the manpulator was moved from the confguraton ( t,,,) to the confguraton ( + δ, + t δ, + t δ, + δ ) and both of the satsfy the eulbrums euatons,.e. J K J (4) and ( + δ) ( J + δj ) K ( + δ ) ( δ) ( J δj) + + (5) where δ J (, ) and δ J (, ) are the dfferentals of the Jacobans due to changes n (, ). Let us also lnearze the geometrc model (5) n the neghborhood of (, ) δt J (, ) δ + J (, ) δ, () After relevant transformaton and neglectng hgh-order small terms, euatons (4), (5) may be rewrtten as J (, ) δ H+,( ) δ H+,( ) δ K δ J (, ) δ H+,( ) δ H+,( ) δ (7) where H, H, H, H,are the Hessan matrces of the scalar functon g(, ). hs allows to apply substtuton for δ and to obtan system of two matrx euatons wth unknowns δ and δ J + k J J J k H δ δt + + J H k J H H k H δ, (8) whch determne stffness model for the case of the loaded eulbrum. Here k ( ) K H. herefore, for a separate knematc chan, the desred stffness matrx K defnng the dsplacement-to-force mappng (4) n the neghborhood of the loaded confguraton can be computed by drect nverson of the matrx n the lefthand sde of (8) and extractng from t the left-upper sub-matrx. Let us note that the matrx (8) can be computed and nverted for any confguraton (ncludng sngular ones). Besdes, the proposed technue takes nto account both elastc deformatons n the vrtual sprngs and unrestrcted knematcs motons due to the passve jonts. In the case of mult-chan manpulator, the desred matrx can be computed n by smple summaton K K., where K corresponds to the -th chan. hs follows from the superposton prncple, snce the total external force correspondng to the end-effector dsplacement δt (the same for each knematc chans) can be expressed as the sum of the partal forces.

5 V. ILLUSRAIVE EXAMPLE (a) Posture A A. Knetostatc model Let us llustrate the proposed technue by stffness analyss of a translatonal manpulator of the Orthoglde famly presented n g.. or ths manpulator, each knematc chan conssts of a foot, a knematc parallelogram wth two axes and two bars, and an end-effector. (g. 2). oot (b) Posture B Bar Passve jonts ool oot 2 3 Bar Axs (c) Posture C End-effector g 2 Chan of the Orthoglde manpulator As follows from a separate study, the rgdty of the parallelogram axes s hgh compared to the bar and the foot. or the remanng elements, the complance matrces (.e. the nverses of the correspondng stffness matrces) were dentfed usng the EA-based methodology and specal accuracy mprovement tools proposed by the authors [5]. Numercal values for these matrces are k oot k Bar (9) (2) or ths analyss, the knematc parallelogram was replaced by a bar element wth double stffness and there were consdered several typcal postures presented n g 3. or such approxmaton, the knematc model of the manpulator chan s expressed as base x ( R a ) V( Root ) y ( ) z ( 2) ( L) V( R ) R( ) ( ) x Lnk z 3 y 4 ool (2) 4 g 3 ypcal postures of the Orthoglde knematc chan where base, ool are constant transformatons matrces (the matrx base ncludes also the foot transformaton) a, oot, Lnk are the vrtual jont coordnates of the actuator, the foot and the lnk respectvely. V (...) s the homogeneous matrx-functon of the vrtual sprngs, whch depends on sx varables and can be descrbed va the multplcaton of elementary transformatons as V( x, y, z, ϕx, ϕy, ϕz) ( ) ( ) ( ) R ( ) R ( ) R ( ) x x y y z z x ϕx y ϕy z ϕz (22) where x, y, z, Rx, Ry, R z are elementary homogeneous transformaton matrces. or ths case study, the Jacoban and Hessan matrces were computed sem-analytcally, va dfferentaton of the model (2). B. Stffness modelng he stffness modelng experments were carred out for four typcal postures presented n g 3 and ABLE I. Durng the stffness modelng, the end-effector of the knematc chan was dsplaced n the range between and 4 mm wth the step. mm, startng from the unloaded confguraton. he statc eulbrum and correspondng force were determned for each dsplacement usng the teratve algorthm presented n Secton 3. Besdes, the stffness matrx was computed for each case. he stffness model was computed seuentally from small to hgh dsplacement, usng prevous state as a startng one for the next step. he force-dsplacement relatonshp for each posture present on the g 4

6 ABLE I CONIGURAIONS O HE MANIPULAOR CHAIN Confguraton Pos. A (g 3, a) Pos. B (g 3, b) / / Pos. C (g 3, c) / / Pos. D / / / / K 3228 K (a) K 8.5 K (c) K 3 K (b) K 2. K g 4 orce-dsplacement relatonshp of the Orthoglde chan: a) Pos. A b) Pos. B c) Pos. C d) Pos. D (d) he obtaned results show that the stffness model of the consdered knematc chan s sgnfcantly nonlnear and essentally depends on both the posture of the manpulator chan and the external force. In partcular, applyng the force about 2 kn leads to the bucklng effect and consderable reducton of the stffness (by 5 tmes, see ABLE I). After the bucklng, even nsgnfcant ncrease of the external force causes very essental deflectons of the knematc chan and correspondng postng errors. Hence, the developed technue allowed detectng some uncommon behavor of the robotc manpulators whch was prevously not reported n robotc lterature. hese phenomena are drectly related to the robot accuracy must be obvously taken nto account durng desgn and analyss. ACKNOWLEDGMEN he work presented n ths paper was partally funded by the Regon Pays de la Lore, rance and by the EU commsson (project NEX). REERENCES [] S. moshenko and J. N. Gooder, heory of elastcty, 3d ed.ew York: McGraw-Hll, 97. [2] G. Pras, W.L. Cleghorn and J.K. Mlls. Dynamc fnte-element analyss of a planar hgh-speed, hgh-precson parallel manpulator wth flexble lnks. Mechansm and Machne heory, vol. 4(7), pp , 25. [3]. Majou, C. Gosseln, P. Wenger, D. Chablat, Parametrc stffness analyss of the Orthoglde, Mechansm and Machne heory, vol. 42, pp [4] C.M. Gosseln, Stffness mappng for parallel manpulators, IEEE ransactons on Robotcs and Automaton, vol., pp , 99. [5] A. Pashkevch, D. Chablat, and P. Wenger, Stffness analyss of 3-d.o.f. overconstraned translatonal parallel manpulators, IEEE Internatonal Conference on Robotcs and Automaton, pp , 28. [] A. Pashkevch, D. Chablat, P. Wenger, Stffness analyss of overconstraned parallel manpulators, Mechansm and Machne heory, vol. 44, pp , 28 [7] Alc, G., Shrnzadeh, B., 25, Enhanced stffness modelng, dentfcaton and characterzaton for robot manpulators, IEEE ransactons on Robotcs vol. 2/4, pp , 25. [8] C. Quennouelle and C. M. Gosseln, Stffness Matrx of Complant Parallel Mechansms, Sprnger Advances n Robot Knematcs: Analyss and Desgn, pp , 28 [9] Y. L and Q. Xu, Stffness analyss for a 3-PUU parallel knematc machne, Mechansm and Machne heory, vol. 43, no. 2, pp. 8-2, 28. [] R. Clavel, DELA, a fast robot wth parallel geometry, Proceedngs, of the 8th Internatonal Symposum of Robotc Manpulators, IR Publcaton, pp. 9, 988. [] D. Chablat and P. Wenger, Archtecture Optmzaton of a 3-DO Parallel Mechansm for Machnng Applcatons, the Orthoglde, IEEE ransactons on Robotcs and Automaton, vol. 9, no. 3, pp. 43-4, 23. [2] R. Vertechy, V. Parent-Castell. Statc and Stffness Analyses of a Class of Over-Constraned Parallel Manpulators wth Legs of ype US and UPS, In: Proceedngs of IEEE Internatonal Conference on Robotcs and Automaton (ICRA), pp. 5 57, 27. [3] O. Company, S. Krut,. Perrot. Modellng and Prelmnary Desgn Issues of a 4-Axs Parallel Machne for Heavy Parts Handlng. Journal of Multbody Dynamcs, 2 (22), pp. -. [4] Steven M. LaValle, Plannng Algorthms, Cambrdge Unversty Press 2 [5] A. Pashkevch, A. Klmchk, D. Chablat, Ph. Wenger, Accuracy Improvement for Stffness Modelng of Parallel Manpulators In: Proceedngs of 42nd CIRP Conference on Manufacturng Systems, June 29 ABLE II INLUENCE O HE EXERNAL LOADING ON HE SINESS O HE MANIPULAOR CHAIN Posture K /m Bucklng Large deformatons K /m cr Δ cr, mm K 2 /m Δ, mm K 3 /m Posture A Posture B Posture C Posture D K s the stffness for the unloaded mode, K s the stffness before the bucklng, K 2 s the stffness after the bucklng, K 3 s the stffness for the large deformatons, cr s the crtcal force for the bucklng, s the force for the large deformatons, Δ cr s the deformaton n the bucklng mode, Δ s the large deformaton value.