Stiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings

Transcription

1 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Alexandr Klmchk, Damen hablat, Anatol Pashkevch o cte ths verson: Alexandr Klmchk, Damen hablat, Anatol Pashkevch. Stffness modelng for perfect and nonperfect parallel manpulators under nternal and external loadngs. Mechansm and Machne heory, Elsever, 214, 79, pp <1.116/j.mechmachtheory >. <hal > HAL Id: hal Submtted on 18 Sep 214 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. he documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfues de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 1 Alexandr Klmchk a,b, 1 Damen hablat b,c, Anatol Pashkevch a,b, a Ecole des Mnes de Nantes, 4 rue Alfred-Kastler, Nantes 4437, France b Insttut de Recherches en ommuncatons et en ybernétue de Nantes, UMR NRS 6597, 1 rue de la Noe, Nantes, France c entre natonal de la recherche scentfue (NRS), France Abstract he paper presents an advanced stffness modelng technue for perfect and non-perfect parallel manpulators under nternal and external loadngs. Partcular attenton s pad to the manpulators composed of non-perfect seral chans, whose geometrcal parameters dffer from the nomnal ones and do not allow to assemble manpulator wthout nternal stresses that consderably affect the stffness propertes and also change the end-effector locaton. In contrast to other works, several types of loadngs are consdered smultaneously: an external force appled to the end-effector, nternal loadngs generated by the assemblng of non-perfect seral chans and external loadngs appled to the ntermedate ponts (auxlary loadng due to the gravty forces and relevant compensator mechansms, etc.). For ths type of manpulators, a non-lnear stffness modelng technue s proposed that allows to take nto account naccuracy n the chans and to aggregate ther stffness models for the case of both small and large deflectons. Advantages of the developed technue and ts ablty to compute and compensate the complance errors caused by the consdered factors are llustrated by an example that deals wth parallel manpulators of the Orthoglde famly. Keywords: Non-lnear stffness modelng, parallel manpulators, complance errors, non-perfect manpulators. 1 Introducton Stffness modelng of robotc manpulator s one of the mportant ssues that allows user to evaluate ts compatblty for certan tasks. It becomes especally crtcal for parallel manpulators, for whch robot manufactures tend to decrease movng masses va reducng the lnk cross-sectons. he latter mproves the robot dynamcs but obvously leads to the deteroraton of the manpulator resstance wth respect to the external forces. In some applcaton areas where hgh precson s reured (robotc based machnng, etc.), the stffness model s n the core of the relevant complance errors compensaton technue and ts precson defnes the fnal product ualty. For these reasons, the problem of the manpulator stffness modelng permanently attracts attenton of the research communty. urrently, there are a number of works that contans essental results on the manpulator modelng under the external loadng. Some of them take nto account very specfc features caused by the nfluence of the second order dervatves [1,2], mpact of passve jonts [3,4] or effect of nternal constrants [5,6]. Other works concentrate the problem of stffness analyss for the manpulators wth nternal preloadng or antagonstc actuatng [7-1]. A number of authors address more general problems arsng n stffness analyss of seral and parallel manpulators [11-14]. here are also ute a few works focusng on partcular archtectures [15-17]. However, some mportant ssues have remaned beyond the scope of research actvtes, they nclude nfluence of nternal loadngs caused by geometrcal errors n over-constraned parallel manpulators and mpact of auxlary loadngs appled to the ntermedate ponts (dfferent from the end-effector). hs paper contrbutes to the enhancement of the exstng stffness modelng methods by focusng on the above mentoned ssues. he remander of ths paper s organzed as follows. Secton 2 presents detaled analyss of related works on the stffness modelng of robotc manpulator. Secton 3 deals wth the stffness modelng of seral chan and proposes eulbrum euatons and a numercal algorthm for computng of the loaded statc eulbrum for the manpulator under nternal and external ladng as well as euatons for computng correspondng stffness matrx and stresses n vrtual jonts. Secton 5 focuses on the stffness modelng of parallel manpulators and presents aggregaton technues for the manpulators wth perfect and non-perfect seral chans under nternal and external loadngs. Secton 6 contans a set of llustratve examples that demonstrate the advantages of developed technue. Secton 7 presents some dscusson concernng lmtatons and possble extensons of the developed method. And fnally, Secton 8 summarzes the man results and contrbutons. 1 orrespondng author. el ; fax ; E-mal address: alexandr.klmchk@mnes-nantes.fr (A. Klmchk).

3 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 2 Related works on the stffness modelng of robotc manpulator o ensure effcent compensaton of the complance errors n the robot-based machnng, an approprate stffness model whch s able to take nto account both changes n manpulator confguraton and nfluence of the external forces/torues s reured. hs secton presents an analytcal revew of exstng approaches n the stffness modelng of robotc manpulators of both seral and parallel archtectures, wth specal attenton to the vrtual jont method provdng a reasonable compromse between the model accuracy and computatonal effcency. In addton, a specal emphass s gven to the stffness modelng n the loaded mode, whch s essental for the consdered applcaton area. 2.1 Problem of stffness modelng and exstng approaches Exstng approaches. Smlar to general structural mechancs, the robot stffness characterzes the manpulator resstance to the deformatons caused by an external force or torue appled to the end-effector [18,19]. Numercally, ths property s usually defned through the stffness matrx K, whch s ncorporated n a lnear relaton between the translatonal/rotatonal dsplacement and statc forces/torues causng ths transton (assumng that all of them are small enough). he nverse of K s usually called the complance matrx and s denoted as k. As t follows from related works, for conservatve systems K s a 6 6 sem-defnte non-negatve symmetrcal matrx but ts structure may be non-dagonal to represent the couplng between the translaton and rotaton [12]. However, n general, for non-conservatve systems and/or some specal parameterzatons of end-effector locaton, n the loaded mode the stffness matrx may be asymmetrcal 2. In stffness modelng of robotc manpulator, because of some specfcty, there are some peculartes n termnology. In partcular, the matrx K s usually referred to as the artesan Stffness Matrx K and t s dstngushed from the Jont-Space Stffness Matrx K that descrbes the relatonshp between the statc forces/torues and correspondng deflectons n the jonts [28]. Both of these stffness matrces can be mapped to each other usng the onservatve ongruency ransformaton [29-31], whch s trval f the external (or nternal) loadng s neglgble. In ths case, the transformaton s entrely defned by the correspondng Jacoban matrx. However, f the loadng s essental, t s descrbed by a more complcated euaton that ncludes both the Jacoban as well as the Jacoban dervatves and the loadng vector [1,5]. Other specfc cases, where the above transformaton s non-trval (non-lnear or even sngular), are related to manpulators wth passve jonts and over-constraned parallel archtectures [3,4,1]. Snce ths work contrbutes to both problems, these ssues wll be consdered n more detal below. In the most general sense, exstng approaches to the manpulator stffness modelng may be roughly dvded nto three man groups: () the Fnte Elements Analyss (FEA), () the Matrx Structural Analyss (MSA), and () the Vrtual Jont Modelng method (VJM). her advantages and dsadvantages are brefly presented below, wth specal emphass to the computatonal complexty and accuracy. Fnte Element Analyss method (FEA). Its basc dea s to decompose the physcal model of the mechancal structure on a number of rather small (fnte) elements and to ntroduce complant relatons between adjacent nods descrbed by relevant stffness matrces. hese fnte elements have a standard shape (pyramds, cubes, etc.) for whch the stffness matrx can be computed analytcally. Usng ths dscretzaton, the statc eulbrum euatons for each node are derved, and they are aggregated n a global matrx expresson defnng relatons between the appled force/torue and node deflectons. hen, the obtaned matrx of rather large sze s nverted and s used to obtan the desred stffness matrx by smple extracton of proper elements. In the modern AD-envronment, the above process s hghly automated and s ntegrated wth 3D-modelng of mechancal structures and mechansms. In partcular, the decomposton nto a set of fnte elements (so called meshng) usually needs defnton of the dscretzaton step and the mesh type only. he latter can be ether lnear or parabolc, whch correspond to pyramds wth 4 and 1 nodes respectvely (wth ether 6 or 12 complance relatons). hen (for ths fnte element model), the AD-based tools provdes both numercal data and convenent vsualzaton defnng the deflectons vectors for each nodes and potental dangerous areas wth hgh stresses. An evdent advantage of the FEA-modelng s ts hgh accuracy that s lmted by the dscretzaton step only. For robotc applcaton t s very attractve, snce the lnks/jonts are modeled wth ts true dmenson and shape [16,32,33]. However, whle ncreasng of the number of fnte elements, the problem of lmted computer memory and the dffculty of the hgh-dmenson matrx nverson become more and more crtcal. Besdes the hgh computatonal efforts, ths matrx nverson generates numerous accumulatve round-off errors, whch reduce accuracy. In robotcs, ths causes rather hgh computatonal expenses for the repeated re-meshng and re-computng, so n ths area the FEA method can be appled for the lnks stffness matrx dentfcaton [34] that s further used n the frame of the VJM technue. Such combnaton allows us to use the advantages of the FEA whle avodng ntensve computatons for dfferent manpulator confguratons. 2 2 In robotc lterature, there s rather ntensve dscussons concernng the symmetry of the stffness matrx n the loaded mode [2-27]. But n ths work, due to the adopted assumptons, the stffness matrx s certanly symmetrcal.

4 A. Klmchk, D. hablat, A. Pashkevch 3 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Matrx Structural Analyss method (MSA). hs method ncorporates the man deas of the FEA but operates wth rather large complant elements such as beams, arcs, cables, etc. [35]. hs obvously leads to the reducton of the computatonal expenses and, n some cases, allows us to obtan an analytcal stffness matrx for the specfc task. Smlar to the FEA-modelng the MSA method gves forces/torues and dsplacements for each node, but here t has a clear physcal nterpretaton (manpulator actve or passve jont), whch can be useful for some tasks [14,36]. For parallel robots, ths method has been developed n works [37,38], where a general technue for stffness modelng of the manpulator wth rgd/flexble lnks and passve jonts was proposed. It has been llustrated by stffness analyss of parallel manpulator of Delta archtecture where the lnks were approxmated by regular beams. he latter causes some doubts n the model accuracy compared to the combnaton of the FEA and VJM technues that are beng developed here. Besdes, ths result was obtaned under the assumpton that the external forces/torues are relatvely small (.e. for the unloaded mode), and t s unlkely that such approach can be enhanced to take nto account partculartes of manpulator behavor n loaded mode. In addton, here there exsts a problem of the stffness matrx computaton for the manpulator sngular confguratons. From a computatonal pont of vew, the MSA method s less complcated than the FEA-based one. In spte of the fact that the MSA stll nvolves matrx operatons of rather hgh-dmenson, t gves a reasonable trade-off between the accuracy and computatonal tme, provded that lnks approxmatons by the beam elements are realstc. It should be also noted that, n ther general formulatons, the FEA and MSA methods are closed: both of them nterpret physcal system as a set of nodes wth mutual flexble connectons. he man dfference s that the MSA operates wth true physcal objects (lke beams, arcs and others), whle the FEA decomposes them nto small fnte elements. So, the MSA can be treated as a specal case of the FEA that has already found ts applcaton n robotcs. From the other sde, f each lnk s appled to the FEA-based stffness matrx dentfcaton technue developed n [34], an advanced combnaton of the MSA and FEA that s sutable for the stffness modelng of arbtrary parallel manpulators (wth numerous nternal loops) can be obtaned. However, t s out of the scope of ths work because of the above mentoned crtcal lmtaton, whch makes the method applcable to the case of the unloaded mode only. Vrtual Jont Modelng method (VJM). he core of ths method s an extenson of the conventonal rgd-body model of the robotc manpulator, where the lnks are treated as rgd but the jonts are assumed to be complant (n order to accumulate all types of exstng flexbltes n the jonts only). Geometrcally, such approxmaton s euvalent to addng to the jonts some auxlary vrtual jonts (wth embedded vrtual sprngs). It s obvous that such lumped presentaton of the manpulator stffness (that n realty s dstrbuted) essentally smplfes the model. So, at present t s the most popular stffness analyss method n robotcs. hs method was frst ntroduced by Salsbury [39] and Gosseln [4], who assumed that the man flexblty sources were concentrated n the actuator jonts. he derved expresson defnng relaton between the jont and artesan stffness matrces (onservatve ongruency ransformaton) became a bass for the manpulator stffness analyss n many research works. Later, these results have been further developed n order to take nto account some specfc geometrcal constrans [5,41] and external loadng [1,2]. Nevertheless, external loadng s assumed small enough to detect any non-lnear effects dscovered n ths work. Due to ts computatonal smplcty, the VJM method has also been successfully appled to the analytcal stffness analyss of a translatonal parallel manpulator [42]. A key ssue of ths method s how to defne the vrtual sprng parameters. At the begnnng, t was assumed that each actuated jont s presented by a sngle one-dmensonal vrtual sprng [11,24,43]. Further, to take nto account the lnks flexbltes, the number of vrtual jonts was ncreased and n each actual actuated or passve jont several translatonal and rotatonal vrtual sprngs were ncluded [42]. he latest developments n ths area operate wth 6-dmensnal vrtual sprngs dentfed usng the FEA-based method [3,34]. hs leads to essental ncreasng of the VJM method accuracy that becomes comparable wth the accuracy of the FEA-based technues, but wth much lower computatonal expenses. he only essental dsadvantage of the VJM method s related to some dffculty n stffness modelng of parallel manpulators. Whle for strctly parallel archtectures, where there are only seral chans between the base and movng platforms, for archtectures wth nternal loops (or wth parallelogram-based lnks) the VJM-base stffness analyss s rather complcated. Nevertheless, takng nto account all advantages and dsadvantages of the FEA, the MSA and the VJM technues, the VJM method looks the most attractve n robotcs. omputatonal complexty. he man benchmark that allows us to evaluate the computatonal complexty of the above descrbed methods s related to the computatonal efforts reured for the matrx nversons nsde of the correspondng algorthms. It should be stressed that the matrx nverson s the most tme consumng operaton for both lnear and non-lnear 3 stffness modelng. Generally, for the matrx nverson of sze n n, t can be defned as O( n ) [44]. For the FEA method n depends on the dscretzaton step and the type of fnte elements used. As t follows from a relevant study, n the case of the parabolc mesh (1 nodes and 12 connectons per a fnte element) the value of n can be approxmately computed as n 3 n, where n s the number of manpulator lnks, L L s the number of fnte elements per lnk (t should be about 1 3 to ensure desred precson). For the MSA method, the upper bound of the above matrx sze n 12n d can be computed

5 A. Klmchk, D. hablat, A. Pashkevch 4 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs va the node number n. Fnally, for the VJM method, the sze of the matrx to be nverted s n 6 n, where n s total number d of the degrees of freedom n the passve jonts. Usng these formulas, t was estmated the stffness model complexty for three types of manpulators (Stewart platform, Delta, Orthoglde). Relevant results are presented n able 1. As t follows from them, the VJM method essentally overcomes FEA and MSA, so t wll be used n ths work as a preferable one. It s presented n more detals below. able 1 omputatonal complexty of exstng stffness modelng methods Stffness modelng method FEA MSA VJM Stewart Platform (6 seral chans, each wth 2 lnks and 5 passve jonts) 16 1 ( n 13) 7 1 ( n 5 6 ) d 3 1 ( n 5 ) L Manpulator archtecture Delta (3 seral chans, each wth 6 lnks and 5 passve jonts) 17 1 ( n 19 ) 8 1 ( n 11 3 ) d 3 1 ( n 5 ) n, n, n are the number of lnks, node ponts and passve jonts (n a sngle chan) respectvely L d 2.2 Vrtual Jont Method n stffness modelng of robots L Orthoglde (3 seral chans, each wth 5 lnks and 4 passve jonts) 17 1 ( n 16 ) 8 1 ( n 13 3 ) d 3 1 ( n 4 ) Snce the VJM-based method s proved to be more computatonally effcent whle provdng acceptable accuracy, let us consder t n more detals. akng nto account some specfctes of the consdered applcaton area, the man attenton wll be pad to the VJM applcablty to the stffness modelng of parallel manpulators (both under-constraned and over-constraned), consderng the mpact of the passve jont, evaluatng the nfluence of the external and nternal forces/torues as well as accuracy mprovement of the stffness model. VJM method background. In the frame of ths method, all types of complance exstng n a real manpulator (both dstrbuted and lumped) are replaced by localzed vrtual sprngs located n ts jonts (able 2). Example of VJM modelng s gven n Fgure 1, where both knematc model of Stewart Platform and ts VJM model are presented. hen, for ths mechansm consstng of rgd lnks and complance jonts, the statc eulbrum euatons s derved and lnearzed n order to obtan artesan stffness matrx, whch n a general case depends on the manpulator posture (confguraton) [1]. Usually t s assumed that the elastc deflectons n vrtual sprngs are relatvely small and lnearzaton s performed n the neghborhood of the eulbrum confguraton correspondng to zero forces and torues (unloaded mode). L Ps Lnk Ac Lnk Ps Ps Lnk Ac Lnk Ps Ps Lnk Ac Lnk Lnk Lnk Ps Ps Lnk Ac Lnk Ps Lnk Lnk Ps Lnk Ac Lnk Ps Platform Ps Lnk Ac Lnk Ps Ps Lnk Ac Lnk Ps Platform Ps Lnk Ac Lnk Ps Ps Lnk Ac Lnk Ps Ps Lnk Ac Lnk Ps Lnk Lnk Ps Lnk Ac Lnk Ps Lnk Lnk Ps Lnk Ac Lnk Ps Lnk Lnk Lnk Lnk a) manpulator archtecture b) VJM model Fgure 1 Knematc model of Stewart Platform and ts VJM model

6 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs able 2 VJM-based modelng of manpulator components Manpulator components VJM-model elements omponent Graphcal presentaton omponents Graphcal presentaton Actuated jont Ac Actuated jont + 6 d.o.f. vrtual sprng Ac Passve jont (non-actuated) Ps Passve jont Ps 5 Rgd lnk Lnk Rgd lnk Lnk Elastc lnk Lnk Rgd lnk + 6 d.o.f. vrtual sprng Lnk Rgd base Base Rgd base Base Elastc base Base Rgd base + 6 d.o.f. vrtual sprng Base han #1 han #1 Rgd platform Platform han #3 Rgd platform Platform han #3 han #2 han #2 Elastc platform han #1 han #2 Platform han #3 Rgd platform + set of 6 d.o.f. vrtual sprng (for each chan) han #1 han #2 Platform han #3 hs technue orgnates from the work of Salsbury [39] who derved a closed-form expresson for the artesan stffness matrx of a seral manpulator assumng that the mechancal elastcty s concentrated n the actuated jonts. he basc euatons have been wrtten as t follows t J ; τ J F; τ K ; (1) where t denotes the end-effector dsplacement n artesan space (both postonal and orentatonal), F s the vector of the external loadng appled to the end-effector (that ncludes both the force and the torue components), s the deflecton n the vrtual jont coordnates caused by ths loadng, τ s the vector of reactons n the elastc jonts, K s the correspondng jont stffness matrx, J s the Jacoban matrx computed wth respect to the elastc jonts. Here, the frst euaton s derved from the manpulator geometrcal model, the second one descrbes the statc eulbrum condton (assumng that the load s not essental), and the thrd euaton presents the lnear elastcty relaton (Hooke's law). After relevant transformatons of (1), the desred lnear relaton between the external loadng F and the end-effector dsplacement t s presented as 1 t J K J F, (2) whch gves artesan stffness matrx 1 K J K J (3) In lterature [26,27,45], the latter may be also presented n a slghtly dfferent form K J K J (4) whch sometmes s referred to as onservatve ongruency ransformaton (), to emphasze that t descrbes mappng from the jont stffness to artesan one, and vce versa. In further works, smlar euatons were obtaned for parallel manpulators assumng that they are not over-constraned and the elastcty s concentrated n the actuator jonts whle the passve jonts are perfect [4]. Other contrbutons to ths area nclude [3,15,22,24,25,46-48], where the VJM method was partally extended. It s also worth mentonng the works where the VJM

7 A. Klmchk, D. hablat, A. Pashkevch 6 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs technue was appled to partcular manpulators, n partcular to the apaman, Orthoglde and H4 robots, Stewart Gough platforms, etc. [42,49-54]. VJM model parameters. In the frst works, t was explctly assumed that the man sources of elastcty are concentrated n actuated jonts. orrespondngly, the lnks were assumed to be rgd and the VJM model ncluded one-dmensonal sprngs only. In other recent works, complance of the lnks has been taken nto account by ntroducng addtonal vrtual jonts descrbng ther longtudnal elastcty [11] or stffness propertes n several drectons [42]. Recent development n ths area use 6-dmensonal vrtual jonts to descrbe elastcty of each lnk [17]. At the begnnng, the stffness parameters of the vrtual jonts descrbng the lnk elastcty (and ncorporated n the matrx K ) were evaluated rather roughly, usng a smplfed representaton of the lnk shape by regular beams. Besdes, t was assumed that all lnear and angular deflectons (compresson/tenson, bendng, torson) are decoupled and are presented by ndependent one-dmensonal sprngs that produce a dagonal stffness matrx of sze 66 for each lnk. Afterwards, ths elastcty model was enhanced by usng complete 66 non-dagonal stffness matrx of the cantlever beam [17,55]. hs allowed takng nto account all types of the translatonal/rotatonal complance and relevant couplng between dfferent deflectons. Other enhancements nclude the lnk approxmaton by several beams, but t gves rather modest mprovement n accuracy. Further advance n ths drecton (applcable to the lnks of complcated shape) led to the FEA-based dentfcaton technue that nvolves vrtual loadng experments n AD envronment and stffness matrx estmaton usng dedcated numercal routnes [34]. he latter essentally ncreased accuracy of the VJM-modelng whle preservng ts hgh computatonal effcency. It s worth mentonng that usual hgh computatonal expenses of the FEA s not a crtcal ssue here, because t s appled only once for each lnk (n contrast to the straghtforward the FEA-modelng for the entre manpulator, whch reures complete re-computng for each manpulator posture). As a result, ths approach allowed the authors to ntegrate accuracy of the FEA-modelng nto the VJM-modelng technue that provdes hgh computatonal effcency. General methodology of ths hybrd approach s presented n Fgure 2. Lnk 1 AD model for the lnk 1... Lnk n AD model for the lnk n FEA-based vrtual experments FEA-based vrtual experments Stffness matrx for the lnk 1... VJM-based stffness model of the manpulator Stffness matrx for the lnk n Fgure 2 Integraton of VJM modelng approach wth FEA-based dentfcaton technue of the stffness model parameters Stffness matrx for manpulators wth passve jonts. Another mportant ssue s related to takng nto account nfluence of the passve jonts, whch are wdely used n parallel manpulators. In the smplest case, when the geometrcal constrants mposed by the manpulator assembly are not redundant, the passve jont coordnates may be just elmnated from the knematc model, allowng us drect applcaton of the Salsbury formula. However, n the case of over-constraned or under-constraned manpulators, the elmnaton technue cannot be used drectly. For seral knematc chans wth passve jonts, the problem has been solved for the general case [56]. In partcular, t was proposed an algorthmc soluton that extends the Salsbury formula and s able to produce the rank-defcent stffness matrces descrbng zero-resstance of the end-effector to certan types of dsplacements, whch do not reure deflectons n the vrtual sprngs (due to presence of passve jonts and/or knematc sngularty of the examned posture). he relevant technue nvolves nverson of dedcated suare matrx of sze ( n 6) ( n 6), whch s composed of the lnks stffness matrces, and knematc Jacobans of both vrtual sprngs and passve jonts (here n s the number of passve jonts). hen, the desred artesan stffness matrx s obtaned by smple extracton of an approprate 6 6 sub-matrx from the computed nverse. orrespondng expresson for stffness matrx computaton can be presented as 1 1 K J K J J J (5)

8 A. Klmchk, D. hablat, A. Pashkevch 7 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs where J s Jacoban related to the vrtual sprngs ( 6 n ), J s Jacoban related to the passve jonts ( 6 n ). he man advantage of ths method s ts computatonal smplcty, snce the number of the vrtual sprngs do not nfluence on the sze of the matrx to be nverted. Besdes, the method does not reure manual elmnaton of the redundant sprng correspondng to the passve jonts, snce ths operaton s nherently ncluded n the numercal algorthm. Because of ts evdent advantages, ths approach found further development [4,57]. In these works, an analytcal expresson for the matrx K has been derved from (5) usng the blockwse nverson and presented as 1 K K J J K J J K K (6) 1 1 where the frst term ( K J K J ) s the stffness matrx of the correspondng seral chan wthout passve jonts and the second term descrbes the mpact of the passve jonts. Another mportant contrbuton n ths area s a recursve procedure that allows user to take nto account the mpact of passve jonts seuentally (one by one or by specfc groups). Relevant expresson has the followng form 1 1 K K K J J K J J K ; 1, 2,... (7) 1 2 where the matrces J J are extracted from the full Jacoban J,,... J J n arbtrary order (column-by-column, or by groups of columns). However, for parallel manpulators wth passve jonts, solutons were obtaned for less general cases. hey nclude pure parallel archtectures where the base and the end platform are connected by strctly seral knematc chans. Here, the total stffness matrx can be presented as the sum of partal matrces correspondng to separate chans (computed usng the above descrbed technue) K ( ) 1 ( ) K J K J J K ; J 1 (8) so the passve jonts are taken nto account easly. Besdes, n ths case the over-constranng of the mechansm does not create addtonal dffcultes. For nstance, for the over-constraned manpulator of Orthoglde famly [58], each of the parallel chans yelds the stffness matrx of rank 4 whle ther aggregaton gves the matrx of full rank 6. But f there exsts a cross-lnkng between the parallel chans (lke n knematc parallelograms, for nstance), ths method can not be appled drectly. For ths case, some nterestng results are presented n [6,41] where the geometrcal constrants were treated n a general way but detaled computatonal technues were not developed. VJM modelng of parallel manpulators: problem of nternal stresses. In spte of the fact that the VJM technue has been orgnally developed for seral manpulators, t can be effcently appled to parallel robots. he basc dea here s to obtan frst the stffness models of each knematc chan separately, and after, to ntegrate them n a unted model correspondng to the parallel manpulator. hs dea was partally mplemented n [3,48], where the manpulator structure was assumed to be strctly parallel (.e. wthout nternal loops) and the knematc chans where assembled n the same end-pont. Under such assumptons, the stffness matrx of the parallel manpulator can be computed va straghtforward summaton of the chan stffness matrces n ( ) K K (9) 1 where the ndex defnes the knematc chans and n s the total number of chans. However, n more general (and practcally mportant) cases where the knematc chans are connected to dfferent ponts of the end-platform, ths formula cannot be appled drectly. Besdes, for parallel manpulators wth parallelogram-based lnks, some essental modfcatons are reured. Other lmtaton of the exstng results devoted to the parallel manpulators s related to the assumpton that the assemblng does not produce any nternal stresses. But n practce, numerous errors are accumulated n seral chans [59] and they cause non-neglgble nternal forces n manpulator jonts (even f the external force appled to the end-effector s eual to zero). he nternal forces may essentally change the manpulator behavor (modfy the stffness matrx, change the end-effector locaton, etc.) and should be obvously taken nto account n the stffness model. However, exstng works gnore ths ssue. Another research ssue s assocated wth stffness modelng of parallel manpulators under loadng, whch has not receved proper attenton yet.

9 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 2.3 Stffness modelng under external and nternal loadngs Manpulator stffness modelng n the loaded mode s a relatvely new research area, whch s worth to be consdered separately. hs subsecton presents analyss of related works and defnes some mportant research problems that wll be n the focus of ths work. Some of them are based on the analogy that can be establshed between robotcs and structural mechancs (t concerns bucklng phenomena, for nstance). ypes of loadngs. Manpulator loadng may be of dfferent nature. For the stffness modelng, t s reasonable to dstngush two man types of loadng, external and nternal ones. he external loadng s caused manly by an nteracton between the robot end-effector and the workpece, whch s processed or transported n the consdered technologcal process [2,26]. Another type of external loadng exsts due to the gravty nfluence on the manpulator lnks, for many heavy manpulators employed n machnng the lnk weght s not neglgble [5,7]. Besdes, to compensate n a certan degree the gravty nfluence, some manpulators nclude specal mechansms generatng external forces/torues n the opposte drecton. It s worth mentonng that the external loadng generated by a technologcal process s always appled to the manpulator end-effector whle others may be appled at ntermedate ponts (at jonts, for nstance). Besdes, the external loadngs caused by gravty have obvous dstrbuted nature. In addton to the above mentoned forces/torues, nternal loadng n some jonts may exsts. For nstance, to elmnate backlash, the jonts may nclude preloaded sprngs, whch generate the force or torue even n standard "mechancal zero" confguraton [9]. hough the nternal forces/torues do not nfluence on the global eulbrum euatons, they may change the eulbrum confguraton and have nfluence on the manpulator stffness propertes. For ths reason, nternal preloadng s used sometmes to mprove the manpulator stffness, especally n the neghborhood of knematc sngulartes. Another case where the nternal loadng exsts by default, s related to over-constraned manpulators that are subject of the so-called antagonstc actuatng [8,6]. Here, redundant actuators generate nternal forces and torues that are eulbrated n the frame of close loops. It s obvous that the both external and nternal loadngs nfluence on the manpulator eulbrum confguraton and, conseuently, may modfy the stffness propertes. So, they must undoubtedly be taken nto account whle developng the stffness model. Stffness matrx for the loaded manpulator. At present, n most of related works the stffness s evaluated n a uas-statc confguraton wth no external or nternal loadng. here s a very lmted number of publcatons that drectly address the loaded mode case (or so-called case of large deflectons ), where n addton to the conventonal elastc stffness n the jonts t s necessary to take nto account the geometrcal stffness arsng due to the change n the manpulator confguraton under the load. Although the exstence of ths addtonal stffness component for elastc structures has been known for a long tme [61], the mportance of ths problem for robotc manpulators has been hghlghted rather recently. he most essental results n ths area were obtaned n [1,2,62] where there are presented both some theoretcal ssues and several case studes for seral and parallel manpulators. Several authors [8,9,6] addressed the problem of stffness analyss for the manpulators wth nternal preloadng or antagonstc actuatng, but n relevant euatons some of the second order knematc dervates were neglected. Usng notaton adopted n ths work and summarzng exstng results [1,2,13], the manpulator stffness matrx for the loaded can be expressed as follows - -1 F K J K K J (1) where K s n n stffness matrx that s nduced by external loadng and s not presented n prevous euaton (3). hs matrx F depends on both the dervatves of the Jacoban J and the loadng vector F. Reured detals concernng computng of K are F gven n [1]. In the frame of the same concept, the manpulator stffness model for the loaded mode was proposed n [5,7], where numerous factors were taken nto account (conventonal external loadng, gravty forces, antagonstc redundant actuaton, etc.). he fnal results for the stffness matrx s presented as - 1 u K J K J (11) where K s a soluton of a non-lnear matrx euaton, whch ncludes the jont stffness and the external/nternal loadngs as u parameters. However, ths approach s rather hard from computatonal pont of vew. Besdes, n ths work the Jacobans and all ther dervatves have been computed not n a "true" eulbrum confguraton (t was unreasonably replaced by unloaded one). For ths reason, snce the eulbrum obvously depends on the loadng magntude, some essental ssues were omtted. As a result, any nonlnear effects have been detected n the stffness behavor of the examned manpulators. Another sgnfcant result, whch should be mentoned here, s related to the stffness modelng of the parallel manpulators wth the cross-lnkage. Frstly ths problem was consdered n [5,7] where all coordnates where decomposed nto two groups (dependent and ndependent ones). But no clear rule for such coordnate splttng has been proposed, besdes the developed 8

10 A. Klmchk, D. hablat, A. Pashkevch 9 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs technue nvolved very non-trval computatons. Further, the cross-lnkage was n the focus of [6,41] where a rather compact expresson for the stffness matrx has been proposed - 1 F I K J K K K J (12) whch, compared to (1), ncludes addtonal matrx K that s nduced by the geometrcal constrants (cross-lnkage). However, I there are stll a number of open uestons concernng the coordnate splttng rule (.e. dvdng them nto dependent and ndependent ones) and computng of the eulbrum confguraton correspondng to the appled loadng. It should be noted that ths problem of (computng the loaded eulbrum) has been omtted n most of the related works. At the same tme, t s clear that the changes n the manpulator confguraton drectly nfluence the Jacobans (and ther dervatves) as well as on the end-effector locaton. For ths reason, computng the Jacobans and Hessans n a tradtonal way (.e. for the unloaded confguraton) may lead to excessvely rough smplfcaton of the stffness model. In partcular, some non-lnear phenomena n manpulator stffness behavor can be hardly detected, whle they should obvously exst from the pont of vew general theory of elastc structures. o our knowledge, the most extended results n ths area has been presented n [1], where the authors proposed a non-lnear stffness model for parallel manpulators wth passve jonts whch take nto account devatons n the manpulator confguraton caused by the external loadng appled to the manpulator end-effector. In the frame of ths concepton, an teratve scheme that allows them to compute a statc eulbrum confguraton has been developed F 1 J J t J J 1 J 1 J K K 1 (13) where the vector t defnes the end-effector dsplacement caused by the external loadng and s preloadng n vrtual sprngs. Snce, ths procedure cannot dstngush a stable eulbrum confguraton from an unstable one, there were proposed a matrx crtera that allows user to check stablty va analyzng matrx propertes o F F o V H K H V o F F o V H H V (14) o o where V, V are the sub-matrces correspondng to the zero sngular values of aggregated Jacoban, J J after applyng to F F F F t SVD-factorzaton (see [1] for detal), H, H, H, H are the Hessans of scalar functon g(, ) F wth respect to vrtual and passve jont coordnates. In accordance wth ths crtera, the manpulator confguraton s stable f (and only f) the matrx (14) s negatve-defnte. he stffness matrx for the manpulator wth passve jonts under the end-pont loadng can be computed as F F F K J k J J J k H F F F F F F J H k J H H k H F F where 1 1 k K H. Among the lmtatons of the proposed approach t should be mentoned that t s sutable for the manpulators under the loadng appled to the end-effector only and wth perfect seral chans. hese lmtatons wll be overcome n ths paper. Nonlnear-behavor of the manpulator under loadng. In mechancs, t has been known snce a long tme that the elastc structures may suddenly change ther confguraton f the loadng exceed some crtcal value. A classcal example s the so-called Euler column that retans ts straght shape untl the loadng. hs effect (bucklng) s well known n structural mechancs, however n robotcs ths aspect has never been studed before. Non-lnear behavor of force-deflecton relaton and possble bucklng effects have been known snce a long tme. However n robotcs, these uestons dd not attract much attenton, manly due to hgh rgdty of commercally avalable robots. But current trends n mechancal desgn of manpulators that are targeted at essental reducton of movng masses motvate relaxng ths assumpton. Hence, non-lnear stffness analyss s also mportant for the robotc manpulator. As t was mentoned before, exstng stffness analyss technues for robots are strctly assumed that loadng cannot change confguratons of an examned manpulator or these changes are neglgble. hs smplfcaton mposes crucal lmtatons for the stffness analyss and, as a result, does not allow us to detect bucklng and other non-lnear phenomena known from general theory of elastc structures. (15)

11 A. Klmchk, D. hablat, A. Pashkevch 1 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Smlar to the classcal mechancs three types of bucklng can appear n a robotc system: bucklng of the lnk, contact bucklng and geometrcal bucklng. Frst type of bucklng s defned by the mechancal propertes of the lnk and easly can be detected by FEA analyss or crtcal loadng for t can be computed va approxmated euatons. Normally these loadngs are unreachable n robots, whle mnmzaton of the lnk cross-sectons can make these lmts reachable. hus t s reasonable to check crtcal loads for the bucklng of elements on the desgn step. he second type of bucklng s caused by the contact of the lnks wth envronment. It can be avoded on the machnng process desgnng stage. he nature of the geometrcal bucklng s closed to the bucklng of the elements, whle several elements should be analysed together. In ths case the crtcal force s defned by the stffness of the lnks and junctons between them. Snce stffness of the juncton may be lower than stffness of lnks, or even n parallel manpulators can be neglgble (for the passve jonts), the crtcal force can be reduced n tmes comparng wth the crtcal loadng of the separate elements. So, nonlnear effects and bucklng can appear n robots and they reure addtonal analyss, however these uestons have been omtted before. In practce, t s mpossble to detect non-lnear effect wthout fndng the loaded eulbrum, whle ths ueston was omtted n prevous works. Besdes, the loadng may potentally lead to multple eulbrums, to bfurcatons of the eulbrums and to statc nstablty of the manpulator confguratons. hese effects are essentaly dangerous for parallel manpulators wch mpose numerous passve jonts. Some aspects of multple-eulbrum problem for robotc manpulators have been examned n the works [63,64] who appled the atastrophe theory for the stablty analyss of the planar parallel manpulators wth several flexural elements under external loadng. However, they dd not propose a general approach for stablty analyss of the manpulator confguratons. herefore, t wll be also n the focus of ths research. 3 Stffness modelng for seral chan wth nternal and external loadngs ypcal examples of the examned knematc chans can be found n the 3-PUU translatonal parallel knematc machne [17], n the Delta parallel robot [65] or n the parallel manpulators of the Orthoglde famly [58] and other manpulators [66]. It s worth mentonng that here a specfc spatal arrangement of under-constraned chans yelds the over-constraned mechansm that posses a hgh structural rgdty wth respect to the external force. In partcular, for Orthoglde, each knematc chan prevents the platform from rotatng around two orthogonal axes and any combnaton of two knematc chans suppresses all possble rotatons of the platform. Hence, the whole set of three knematc chans produces a non-sngular stffness matrx whle for each separate chan the stffness matrx s sngular. hs motvated the development of dedcated stffness analyss technues that are presented below. 3.1 Problem statement Let us consder a general seral knematc chan, whch conssts of a fxed Base, a number of flexble actuated jonts Ac, a seral chan of flexble Lnks, a number of passve jonts Ps and a movng Platform at the end of the chan (Fgure 3). It s assumed that all lnks are separated by jonts (actuated or passve, rotatonal or translatonal) and the jont type order s arbtrary. Besdes, t s admtted that some lnks may be separated by actuated and passve jonts smultaneously. Such archtecture can be found n most of the parallel manpulators where several smlar knematc chans are connected to the same base and the platform n a dfferent way (wth the rotaton of 9 or 12, for nstance), n order to elmnate the redundancy caused by the passve jonts. It s obvous that such knematc chans are statcally under-constraned and ther stffness analyss cannot be performed by the drect applcaton of the standard methods. Base platform (rgd) Ac Ps Lnk Ps... Ac Ps Lnk G 1 G 2 Fgure 3 G n-1 Ps Knematc chan Moble platform (rgd) G n F 6-d.o.f. sprng 6-d.o.f. sprng... Ac Ps Lnk Ps Lnk Ps a) knematc model b) VJM model General structure of knematc chan wth auxlary loadng and ts VJM model In order to evaluate the stffness of the consdered seral chan, let us apply a modfcaton of the vrtual jont method (VJM), whch s based on the lump modellng approach [39,4]. Accordng to ths approach, the orgnal rgd model should be extended by addng vrtual jonts (localzed sprngs), whch descrbe elastc deformatons of the lnks. Besdes, vrtual sprngs are ncluded n the actuatng jonts, to take nto account the stffness of the control loop. Under such assumptons, the knematc chan can be descrbed by the followng seral structure: (a) G 1 Lnk G 2 6-d.o.f. sprng a rgd lnk between the manpulator base and the frst actuatng jont descrbed by the constant homogenous transformaton matrx ; Base Lnk G n F

12 A. Klmchk, D. hablat, A. Pashkevch 11 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs (b) the 6-d.o.f. actuatng jonts defnng three translatonal and three rotatonal actuator coordnates, whch are a a a a a a descrbed by the homogenous matrx functon ( ) where (,,,,, ) are the vrtual 3D a a x y z φx φy φz sprng coordnates; (c) (d) (e) the 6-d.o.f. passve jonts defnng three translatonal and three rotatonal passve jonts coordnates, whch are descrbed by the homogenous matrx functon ( ) where (,,,,, ) are the passve jont 3D p p x y z φx φy φz coordnates; the rgd lnks, whch are descrbed by the constant homogenous transformaton matrx ; Lnk a 6-d.o.f. vrtual jont defnng three translatonal and three rotatonal lnk-sprngs, whch are descrbed by the homogenous matrx functon ( ), where (,,,,, ), (,, ) and (,, ) 3D Lnk Lnk x y z φx φy φz x y z φx φy φz correspond to the elementary translatons and rotatons respectvely; (f) a rgd lnk from the last lnk to the end-effector, descrbed by the homogenous matrx transformaton. ool In the frame of these notatons, the fnal expresson defnng the end-effector locaton subject to varatons of all jont coordnates of a sngle knematc chan may be wrtten as the product of the followng homogenous matrces (16) Base 3D a 3D p Lnk 3D Lnk 3D p ool where the components, (...),, may be factorzed wth respect to the terms ncludng the jont varables, n Base 3D Lnk ool order to smplfy computng of the dervatves (Jacoban and Hessan). hs expresson ncludes both tradtonal geometrc varables (passve and actve jont coordnates) and stffness varables (vrtual jont coordnates). he explct poston and orentaton of the end-effector can by extracted from the matrx n a standard way [19], so fnally the knematc model can be rewrtten as the vector functon t g (, ) (17) where the vector t ( p, φ ) ncludes the poston p ( x, y, z ) and orentaton φ (,, ) of the end-platform, the x y z vector (,,..., ) contans all passve jont coordnates, the vector (,,..., ) collects all vrtual jont 1 2 n 1 2 n coordnates, n s the number of the passve jons, n s the number of the vrtual jonts. Several examples of prsmatc passve and actuated jonts are presented n Fgure 4a-c, some other types of jonts have been llustrated n [67-7]. Such jonts nclude nternal sprngs, as such ther statcs s descrbed by the followng expresson K (18) where s the torue/force caused by the devaton of the jont coordnate from ts unloaded ( zero ) value, and coeffcent K defnes the sprng stffness. For the purpose of generalty, let us ntroduce smlar zero values for the vrtual sprngs that descrbed flexblty of the lnks (obvously they are eual to zero for ths subset of ). hs allows us to defne vector of the same sze as and to present the statc euatons correspondng to all varables (correspondng to perfect and preloaded passve jonts, vrtual sprngs of lnks and actuators) n general form τ K ( ), τ (19) M (a) Passve jont wth hard end-constrans Fgure 4 (b) Actuated jont wth elastc transmsson Examples of prsmatc passve and actuated jonts (a) Passve jont wth soft end-constrans

13 A. Klmchk, D. hablat, A. Pashkevch 12 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Here τ, τ are the generalzed torue/force n jonts correspondng to the varables and ; the matrx K collects stffness coeffcents of all sprngs of the knematc chan. In the frame of ths paper t s assumed that the nternal loadng (whch may change the robot stffness propertes) may arse from two reasons: () due to addtonal elastc elements ntroduced by the desgner n order to mprove the manpulator propertes; () because of nternal stresses n lnks/jonts caused by over-constraned archtecture of the consdered manpulators. It s clear that here the total sum of all nternal forces/torues s eual to zero (n contrast to the external loadng studed n most of related works); nevertheless the nternal loadng nfluence on the manpulator stffness matrx may be essental. It s worth mentonng that n the case wthout nternal preloadng, the vector descrbes only flexblty of manpulator lnks/actuators that are presented by vrtual sprngs, whle vector collects entre set of passve jont coordnates. In contrast, here, the passve jont coordnates are dvded nto two subsets: () perfect passve jonts ncluded n, and () preloaded passve jonts ncluded n together wth tradtonal vrtual sprngs. Besdes, f a passve jont ncludes a nonlnear sprng (see Fgure 4c), the correspondng jont varable may be ncluded ether n or, dependng on the current confguraton of the manpulator. However, for each confguraton, ths assgnment s strctly unue. It s assumed that the desred stffness model of seral chan s defned by a non-lnear relaton F f (Δ t ), (2) where f (...) s a so-called force-deflectons functon that assocates a deflecton Δ t wth an external force F that causes deformatons. It s worth mentonng that the functon f (...) can be determned even for the sngular confguratons (or redundant knematcs) whle the nverse statement s not generally true. Hence, enhanced stffness analyss must nclude the computaton of ths functon and the detaled analyss of ts sngulartes that may provoke varous nonlnear phenomena (such as bucklng). In the unloaded case, ths functon s usually defned through the stffness matrx K, whch descrbes the lnear relaton F K (, ) Δ t between small sx-dmensonal translatonal/rotatonal dsplacements Δt, and the external forces/torues F causng ths transton. Here, t s assumed that Δt ncludes three postonal components ( x, y, z ) descrbng the dsplacement n artesan space and three angular components (Δ, Δ, Δ ) that descrbe the end-platform rotaton around x y z the artesan axes, whle the vectors, correspond to the manpulator eulbrum confguraton for whch the loadngs (both nternal and external) are eual to zero. However, for the loaded mode, smlar lnear relaton s defned n the neghborhood of another statc eulbrum, whch corresponds to a dfferent manpulator confguraton (, ), that s modfed by external forces/torues F. Respectvely, n ths case, the stffness model descrbes the relaton between the ncrements of the force F and the poston t F F K (, ) t (21) where Δ and Δ denote the new confguraton of the manpulator, and Δ, Δ are the devatons n the coordnates, respectvely. For stffness modelng of seral knematc chan wth auxlary loadng let us assume that the seral chan has the addtonal external loadngs appled to the nternal node ponts (Fgure 3). hese loadngs can be caused by gravty forces (generally they are dstrbuted, but n practce they can be approxmated by localzed ones) and/or gravty compensators. hese forces wll be denoted as G, where j j 1,..., n s the node number n the seral chan startng from the fx base (here, j n corresponds to the end-pont). It should be noted that for computatonal convenence, t s assumed that the end pont loadng conssts of two components G and F of dfferent nature. n It s evdent that n general the auxlary forces descrbed by the functons G depend on the manpulator confguraton. So, let us assume that they are G G (, ), (22) j j In contrast, for the external force F, t s assumed that there s no drect relaton wth the manpulator confguraton. For the seral chans wth the auxlary loadngs t s also reured to extend the geometrcal model. In partcular, n addton to the euaton (17) defnng the end-pont locaton, t s necessary to ntroduce the addtonal functons t g (, ), j 1,..., n (23) j j defnng locatons of the nodes. It s worth mentonng that for the seral chan, the poston t depends on a sub-set of the jont j coordnates (correspondng to the passve and vrtual jonts located between the base and the j-th node), but for the purpose of analytcal smplcty let us use the whole set of the jont coordnates (, ) as the arguments of the functons g (...). Usng these assumptons, the problem of stffness modelng of seral chans wth auxlary loadngs can be splt n the followng sub-problems: (a) dervng the statc eulbrum euatons for the chan wth auxlary loadngs; (b) computng

14 A. Klmchk, D. hablat, A. Pashkevch 13 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs full-scale force-deflectons relaton for the end-pont and ntermedate nodes; (c) lnearzaton of the relevant force-deflecton relatons n the neghborhood of the eulbrum and computng correspondng stffness matrx. Let us focus on these sub-problems. 3.2 Statc eulbrum euatons for seral chan wth auxlary loadngs o obtan a desred stffness model, let us derve frst the statc eulbrum euatons. In the frame of ths work, the noton of statc eulbrum of seral chan s referred to the confguraton (defned by a set of the actuated, vrtual and passve jont coordnates) that depends on external/auxlary loadngs, whch ensure zero sums of forces/torues for each lnk separately. Let us apply the prncple of the vrtual work and assume that the knematc chan under external loadngs F and G G... G 1 n has the confguraton, and the locatons of the end-pont and the nodes are t g (, ) and t g (, ), j 1, n respectvely. j j Accordng to the prncple of vrtual work, the work of external forces G, caused by dsplacement of the vrtual sprngs F s eual to the work of nternal forces τ n G t F t τ (24) j j j 1 where the vrtual dsplacements t j can be computed from the lnearzed geometrcal model derved from (23) ( ) ( ) j j t J J, j 1.. n, (25) j whch ncludes the Jacoban matrces ( j J ) g J ( ) g j j j, ;, (26) wth respect to the vrtual and passve jont coordnates respectvely. Substtutng (25) to (24) we can obtan the euaton n ( j ) ( j ) ( n ) ( n ) G J G J F J F J τ (27) j j j 1 whch has to be satsfed for any varaton of,. It means that the terms regroupng the varables, have the coeffcents eual to zero, hence the force-balance euatons can be wrtten as n n ( j ) ( n ) ( j ) ( n ) ; j j j1 j1 τ J G J F J G J F. (28) Also, these euatons can be re-wrtten n block-matrx form as where (G) (F) (G) (F) ; τ J G J F J G J F (29) (F) ( n ) ( F ) ( n ) (G) (1) ( n ) (G ) (1) ( n ) ; ;... ;... ;... 1 n J J J J J J J J J J G G G (3) Fnally, takng nto account the vrtual sprng reacton τ K, where dag,..., eulbrum euatons can be presented as (G ) (F) J G J F K (G ) (F) J G J F K K K, the desred statc 1 n (G) It should be noted that compared to the case of end-effector loadng only [1], here there are two addtonal terms J G and (G) J G that take nto account the nfluence of the auxlary loadng G. Further, these euatons wll be used for computng the statc eulbrum confguraton and correspondng artesan stffness matrx. (31)

15 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 3.3 Eulbrum confguraton for seral chan wth auxlary loadngs o obtan a relaton between the external loadng F and nternal coordnates of the knematc chan (, ) correspondng to the statc eulbrum, euatons (31) should be solved ether for the dfferent gven values of F or for the dfferent gven values of t. Based on ths data, the desred value of the end-pont dsplacement can be computed straghtforwardly, usng geometrc euaton (17). In prevous works, ths ssue was usually gnored and the lnearzaton was performed n the neghborhood of the unloaded confguraton assumng that the external load s small enough. It s obvous that the latter essentally lmts relevant results and does not allow detectng non-lnear effects such as bucklng. From a mathematcal pont of vew, the problem s reduced to solvng of a system of the non-lnear statc eulbrum euatons that may produce unue or non-unue, stable or unstable solutons. For computatonal reasons, let us consder the dual problem that deals wth determnng the external force F and the manpulator eulbrum confguraton (, ) that corresponds to the end-effector locaton t takng nto account nternal preloadng n the jonts and auxlary loadng G,. Let us solve statc eulbrum euatons wth respect to the manpulator confguraton, and external loadng F for gven end-effector poston t g, and functon of auxlary-loadngs G, (G ) (F) (G ) (F) ; K J G J F J G J F, ;, t g G G Snce ths system of euatons usually has no analytcal soluton, an teratve numercal technue can be appled. Smlar to the prevous Secton, the knematc euatons may be lnearzed n the neghborhood of the current confguraton (, ) (F) (F) t g, J, J, ; (33) (G) (F) (G) (F) where the subscrpt '' ndcates the teraton number and the changes n Jacobans J, J, J, J and the auxlary loadngs G, are assumed to be neglgble from teraton to teraton. orrespondngly, the statc eulbrum euatons n the neghborhood of (, ) may be rewrtten as,,, 1 1 (G ) (F) J G J F K ),,, (G ) (F 1 J G J F 14 (32). (34) hus, combnng (33) and (34), the teratve algorthm for computng of the statc eulbrum confguraton for the gven end-effector locaton can be presented as 1 (F) (F) (F) (F) F,,,,, 1 1 J J t g J J (F) (G) 1,, J J G (F) (G) 1 J, K J, G K (35) where G G (, ) o reduce the sze of a matrx to be nverted, the above system can be slghtly smplfed. In partcular, applyng the same 1 (G) (F) approach as n [1] (but here based on analytcal expresson for K ( J G J F ) ), the unknown varables can be separated n two groups ( F, ) and. hs yelds the teratve scheme 1 (F) 1 (F) (F) F,,, 1 J K J J (F) 1 J, (F) (F) (F) 1 (G),,,,, (G) J, G t g J J J K J G 1 1 (G ) (F), 1 K J G J, F 1 (36) he latter s more convenent computatonally, snce t reures the nverson of a lower dmenson matrx n 6 n 6 nstead of n m 6 n m 6, where n, m are the szes of the vectors and respectvely. For nstance, for the knematc chans of the Orthoglde manpulator (see applcaton example n Secton 5), the expresson (35) reures the nverson 1 of matrx, whle teratve scheme (36) needs the nverson of 1 1 matrx only. It should be mentoned that K s computed only once, outsde of the teratve loop.

16 A. Klmchk, D. hablat, A. Pashkevch 15 Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Smlar to the other teratve schemes, convergence of ths algorthm hghly depends on the startng pont. However, due to the physcal nature of the consdered problem, t s possble to start teratons from the non-loaded confguraton (, ). Besdes, t s useful to modfy the target pont for each teraton n accordance wth the expresson t t 1 t usng scalar varable that s monotoncally ncreasng from up to 1. Another approach can be used for computng the force-deflecton curve. Here, the startng pont can be taken from prevously computed loaded confguraton correspondng to another value of deflecton that s very close to the target one. For typcal values of deformatons, the proposed teratve procedure convergences n 3-5 teratons f the confguraton s stable. In the smulaton studes, the convergence has been evaluated by the weghted sum of resdual norms correspondng to euatons (36) and the algorthm stopped when ths crteron acheved the prescrbed value. However, some computatonal dffcultes may arse n the case of bucklng or n the area of multple eulbrums, where the convergence problem becomes rather crtcal and hghly depends on an ntal pont. Here, the number of teratons ncreases sgnfcantly and the computatonal tme becomes non-neglgble. o overcome these dffcultes, t s proposed to modfy the developed teratve scheme and to repeat the computatons several tmes, wth slghtly modfed ntal ponts that are obtaned by addng small random nose to,. Another opton s to add small dsturbances to, at each teraton. hese technues were used n the case studes presented n Secton 5. he proposed teratve scheme can also be slghtly modfed to solve the orgnal problem,.e. computng the eulbrum confguraton correspondng to gven external loadng F (nstead of gven t ). In ths case, expressons (35), (36) are used n the nternal loop, whle the desred algorthm s supplemented by an external loop, whch provdes teratve searchng for t correspondng to the gven F 1 1 t t K F F (37) where t, F and K are the locaton, the loadng and the stffness matrx at the -th teraton respectvely. It s worth mentonng that the dual problem s meanngful only f the stffness matrx K s non-sngular. It s obvous that for a separate seral chan wth passve jonts the matrx K s always sngular, whle for an ndustral seral manpulator t s usually non-sngular (the same as for a parallel manpulator due to specfc assemblng of knematc chans). On the other hand, the dual problem consdered n ths Secton (.e. computng F correspondng to t ) s always physcally meanngful and has at least one soluton. Another problem related to computng eulbrum confguraton that to be consdered, s the stablty of the manpulator confguraton under the loadng. hs problem has been descrbed n detals n [1,71] and can be appled drectly for the consdered case snce the matrx crteron (14) depends on the parameters that defne the end-effector coordnates only; the mpact of loadngs appled to the ntermedate ponts n ths case s taken nto account on the step of the eulbrum confguraton computng. Hence, the proposed algorthm allows us to compute statc eulbrum confguraton for the seral chans wth passve jonts and all types of loadngs (nternal preloadng, external loadngs appled to any pont of the manpulator and loadng from the technologcal process appled to the end-effector). 3.4 Stffness matrx for seral chan wth auxlary loadngs he prevous sub-secton allows us to obtan the non-lnear relaton between elastc deflectons Δ t and external loadng F. Snce common engneerng practce operates wth the stffness matrx, let us lnearze ths relaton n the neghborhood of the eulbrum. Followng the vrtual work technue, let us assume that the external force and the end-effector locaton are ncremented by some small values F, t n the neghborhood of current eulbrum confguraton. Let us also assume that a new confguraton also satsfes the eulbrum condtons. Hence, t s necessary to consder the two eulbrums correspondng to the manpulator state varables ( F,,, t ) and ( F F,,, t t ) smultaneously. he relevant statc eulbrum euatons may be wrtten as and t g, (G ) (F) K J G J F (G ) (F) J G J F (38)

17 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs t t g, (G) (G) (F) (F) K J J G G J J F F (G) (G) (F) (F) J J G G J J F F 16 (39) where the varables t, F, G, K,,, are assumed to be known. After lnearzaton of the functon g(, ) n the neghborhood of loaded eulbrum, the system (38), (39) s reduced to three euatons (F) (F) t J J (G) (G) (F) (F) K J G J G J F J F (G) (G) (F) (F) J G J G J F J F (4) whch defne the desred lnear relatons between t and F,, that are expressed by the stffness matrces. In ths system, small varatons of Jacobans may be expressed va the second order dervatves K K, K, where (F) (F) (F) (F) (F) (F) J H H ; J H H ; (G) (G) (G) (G) (G) (G) J H H ; J H H ; n 2 n 2 (G ) (G ) 2 2 j1 j1 j j j j H g (, ) G ; H g (, ) G ; n 2 n 2 (G ) (G ) j 1 j 1 ( j j j j H g, ) G ; H g (, ) G ; 2 2 (F) (F) H 2 g (, ) F ; H 2 g (, ) F ; 2 2 (F) ( (, ) ; F) ( F H g F H g, ) ; (41). (42) Also, the auxlary loadng G may be computed va the frst order dervatves as G G G (43) Furthermore, let us ntroduce the addtonal notaton (F) (G ) (G ) (G ) (F) (G ) H H H J G ; H H H J G ; (G ) (F) (G ) (G ) (F) (G ) H H H J G ; H H H J G, (44) whch allows us to present the system (4) n the form (F) (F) t J J F (F) J H H (F) J H K H (45) he latter gves a straghtforward numercal technue for computng the desred stffness matrx: drect nverson of the matrx n the left-hand sde of (45) and extractng from t the upper-left sub-matrx of sze 66. Smlarly, the matrces defnng lnear relatons between the end-effector ncrement t and the ncrements of the jont varables, can be computed,.e.: where F K t; K t; K t (46)

18 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs (F) (F) K * * J J (F) K J H H (F) K J H K H 1 17 (47) It s worth mentonng that the nternal preloadng (expressed by the varable ) s not ncluded n the latter expresson n the explct way, but t drectly nfluences the varables (, ) descrbng the eulbrum confguraton and correspondng Jacobans and Hessans, whch are elements of (47). Besdes, n contrast to prevous works, here t s possble to obtan supplementary matrces K, K that gve addtonal measures of the manpulator stffness whch evaluate senstvty of the jont coordnates (, ) wth respect to the external loadng. In the case when the matrx nverse (47) s computatonally hard, the varable can be elmnated analytcally, usng F (F) F F F correspondng statc euaton: k J F k H, where k denotes the modfed jont complance matrx F F 1 k ( K H ). hs leads to a reduced system of matrx euatons wth unknowns F and (F) F (F) (F) (F) F J k J J J k H F t (F) F (F) F J H k J H H k H (48) that may be treated n the smlar way,.e. the desred stffness matrx s also obtaned by drect nverson of the matrx n the left-hand sde of (48) and extractng from t the upper-left sub-matrx of sze 66: (F) F (F) (F) (F) F K J k J J J k H (F) F (F) F K J H k J H H k H 1 (49) Smlar to prevous subsecton, ths approach allows us to reduce the dmenson of the nverted matrx from n m 6 n m 6 to n 6 n 6, that n the case of Orthoglde corresponds to and 1 1 respectvely. It s worth mentonng that the structure of the latter matrx s smlar to the one obtaned for the unloaded manpulator and for the manpulator under end-pont loadng only [1] and dffers by Hessans that take nto account the nfluence of the external load. It should also be noted that, because of the presence of the passve jonts, the stffness matrx of a separate seral knematc chan s always sngular, but aggregaton of all the chans for a parallel manpulator produces a non-sngular stffness matrx. Further smplfcaton of (49) can be obtaned by applyng the block matrx nverson technue of Frobenus [72] that yelds the followng expressons ( F ) ( F ) F F K K K J J k H K (5) ( F ) F 1 where the frst term K ( J k J ) exactly corresponds to the classcal formula defnng stffness of the knematc chan wthout passve jonts n the loaded mode [1,2] and 1 F F F F F F F ( ) F F F F ( F ) K H H k H J H k J K J J k H J H k J K (51) Smlarly, the matrx K can be expressed analytcally as F F F K k J K k H K (52) Hence, the developed technue allows us to compute the statc eulbrum confguraton and artesan stffness matrx for seral chans wth external and nternal loadng appled to the end-effector ad to the node-ponts (auxlary loadng). It allows us to compute values of the nternal varables correspondng to the eulbrum, to obtan the non-lnear force-deflecton relaton and to compute the related stffness matrces. 4 Stffness model of parallel manpulators wth nternal and external loadngs he non-lnear stffness modelng technue proposed n Sectons 3 deals wth separate knematc chans. In order to be appled to the parallel manpulators, t should be extended by approprate stffness model aggregaton routnes. Hence, t s reasonable to propose the method of aggregaton of the elastostatc models of separate knematc chans to the stffness model of the parallel manpulator. In general, these routnes have to be applcable for both perfect and non-perfect knematc chans. Besdes, t s reured to develop numercal algorthms for computng both drect and nverse force-deflecton relatons that are referred below to as non-lnear stffness and complance models respectvely.

19 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 4.1 Stffness model aggregaton technue for perfect seral chans Let us assume that a parallel manpulator may be presented as a strctly parallel system of the actuated seral legs connectng the base and the end-platform (Fgure 5) [66]. Usng the methodology descrbed n the prevous sectons and applyng t to each ( ) leg, a set of m artesan stffness matrces K expressed wth respect to the same coordnate system but correspondng to dfferent platform ponts can be computed. If ntally the chan stffness matrces have been computed n local coordnate systems, ther transformaton s performed n a standard way [19], as 18 K R R glob R loc R K (53) where R s a 3 3 rotaton matrx descrbng orentaton of the local coordnate system wth respect to the global one. ( ) o aggregate these matrces K, they must be also re-computed wth respect to the same reference pont of the platform. Assumng that the platform s rgd enough (as compared to the complance of the legs), ths converson can be performed by extendng the legs by a vrtual rgd lnk connectng the end-pont of the leg and the reference pont of the platform (see Fgure 5 where these extensons are defned by the vectors v ). (a) BP 1 han #1 u 1 EP 1 han #2 EP 2 v F u 2 2 RP M v v han 1 3 EP 3 u #3 3 Platform BP 2 BP 3 (b) han #1 (1) K EP 2 EP 1 han #2 RP v 1 v 2 (2) K v 3 M F EP 3 (3) K han #3 (c) han #2 EP 2 v 2 RP EP 1 v 1 han #1 K v 3 F M EP 3 han #3 Fgure 5 ypcal parallel manpulator (a) and transformatons of ts VJM models (b, c) (EP s th end-pont, BP s th base pont, RP s the reference pont, u E P B P v RP EP ), After such extenson, an euvalent stffness matrx of the leg may be expressed usng relevant expresson for a usual seral ( ) ( ) ( ) 1 chan,.e. as ( ) J K J, where the Jacoban J defnes dfferental relaton between the coordnates of the -th vrtual v v v sprng and the reference frame of the end-platform. Hence, the fnal expresson for the stffness matrx of the consdered parallel manpulator can be wrtten as m 1 ( m ) ( ) ( ) ( ) С v С v 1 K J K J (54) where m s the number of seral knematc chans n the manpulator archtecture. As a result, expresson (54) allows us to compute artesan stffness matrx for the parallel manpulator based on stffness ( ) matrces for seral chans and transformaton Jacobans J, whch defne geometrcal mappng between end-ponts of seral v chans and reference pont frame (end-effector). Moreover, t s mplctly assumed here that all stffness matrces (both for the seral chans and for the whole manpulator) are expressed n the same global coordnate system, otherwse they should be recomputed to reuste coordnates. Hence, the axes of all vrtual sprngs are parallel to the axes x, y, z of ths system. hs allows ( ) to evaluate Jacobans J and ther nverses from the geometry of the end-platform analytcally v J I ( v ) I ( v ) ( ) 3 ( ) 1 3 v, J v I 3 66 I 3 66 (55) where I s 3 3 dentty matrx, and 3 v s a skew-symmetrc matrx correspondng to the vector v : v v z y ( v ) v v (56) z x v v y x herefore, expresson (54) represents explct aggregaton of the leg stffness matrces wth respect to any gven reference ( ) pont of the platform. It s worth mentonng that n practce, the matrces K are always sngular whle the aggregaton usually produces non-sngular matrx. Relevant examples are presented n the followng sub-secton.

20 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 4.2 Stffness model aggregaton technue for non-perfect seral chans (under nternal loadngs) In the prevous Secton, t was mplctly assumed that knematc chans of the parallel manpulator are "perfect",.e. ther geometrcal parameters are strctly nomnal and ther end-frames can be algned and matched wthout addtonal efforts and/or nternal stresses, whle assemblng. However, n practce, the knematc chan geometry may dffer from the nomnal one (.e. be "non-perfect"), and the assemblng causes nternal stresses and shftng of the end-ponts locaton. hus, let us extend the above aggregaton procedure to the case of non-perfect seral chans and develop a technue, whch s able to evaluate deflectons and nternal forces/torues caused by geometrcal errors n the chans. Let us consder a parallel manpulator, whch may be presented as a strctly parallel system of the actuated seral legs connectng the fxed base and moble end-platform [66]. Usng the methodology descrbed n the prevous sectons, each -th leg (seral chan) may be characterzed by the geometrcal functon t g (, ), where t defnes the end-frame locaton and, are passve and vrtual jont coordnates respectvely, whch defne knematc chan confguraton. For perfect knematc chans, the jont coordnates, are computed usng nomnal geometrcal model of the leg g (, ), for the gven end-platform locaton t. hese notatons are llustrated by Fgure 6a where A correspond to the end-ponts of the perfect chans and O s the platform reference pont, the vectors AO are denoted as v. Usng the above presented technue, t s possble to compute the artesan stffness matrces of the chans and express them wth respect to the same reference pont O. Let us denote ( ) ths matrx as K. 19 Fgure 6 ransformaton of characterstc ponts of seral chans n assemblng of non-perfect parallel manpulator; ( A, A - end-pont locatons of seral chan before assemblng for perfect and non-perfect manpulators respectvely, A - end-pont locaton of seral chan after assemblng for non-perfect manpulator) For the non-perfect chan before assemblng, smlar confguratons, produce slghtly dfferent end-pont locatons of the chans t g (, ), whch are denoted as A n Fgure 6a. orrespondently, assumng that the platform s rgd enough, the ponts A can be mapped to B that dffer from O by. Hence, geometrcal errors do not allow assemblng parallel manpulator at the orgnal reference pont. o assemble all chans n the same reference pont, t s reured to apply addtonal efforts. Geometrcally, t leads to relocaton of the frames correspondng to the ponts B to a new poston B, and relocatons of the pont A to A that s defned by the vector Δ t. In Fgure 6b, relevant reference pont of the platform s denoted as O. Hence, the stffness model aggregaton problem s reduced to computng of a new end-platform locaton t t for whch end-frames of all knematc chans are algned and matched. onseuently, a manpulator wll occupy the most advantageous confguraton wth respect to the potental energy of the elastc elements. o compute the end-platform deflecton t, let us assume that the geometrcal errors are small enough and correspondng ( ) shftng of the chan end-ponts and end-effector do not change artesan stffness matrces K or ther nfluences are С ( ) neglgble. So, the stffness matrces of the seral chans K are the same at the ponts O, B and O and computed for the С nomnal confguratons,. Let us also assume that the pont O s shfted from O by Δp and the moble platform orentatons for the parallel manpulator wth perfect and non-perfect seral chans dffer by φ. hs allows us to express the potental energy of the parallel manpulator wth geometrcal errors n knematc chans as

21 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs E m ( ) t K t (57) where t ( p, φ ) s dsplacement (poston and orentaton) of the reference pont. It s obvous that after assemblng, the consdered mechancal system wll occupy the most advantageous confguraton wth respect to the potental energy,.e. E mn. It means that the desred vector t can be found from the expresson E t m K t 1 ( ) that yelds the followng lnear euaton m 1 m ( ) ( ) 1 t K t K (59) whch allows us to evaluate the end-platform deflecton 1 m ( ) ( ) 1 m t K K (6) 1 and the end-platform locaton after assemblng t t t (61) For each separate knematc chan, the end-frame deflectons due to assemblng can be expressed as 1 m m ( ) ( ) 1 1 Δt t K K (62) hs allows us to compute the loadng for each knematc chan appled to the end-pont (due to nteracton wth other non-perfect chans) ( ) F K Δt (63) and correspondng loadngs n the vrtual jonts ( ) ( ) ( ) ( ) J J τ ( ) τ F K Δt (64) m It s worth mentonng that here F, snce there s no external loadng appled to the platform reference pont after the 1 assemblng. Besdes, t s possble to compute relevant deflectons of the vrtual and passve jont coordnates of the chans ( ) ( ) ; K Δt K Δt (65) caused by the assemblng. hus, the above expressons allow us to evaluate the end-platform deflecton and nternal forces/torues caused by assemblng of knematc chans wth geometrcal errors. However, the total manpulator artesan stffness matrx K s assumed to be the same as n the case of perfect seral chans, snce the geometrcal errors are assumed to be small enough. It should be mentoned that the dfference n the potental energy due to the geometrcal errors of seral chans s the second order of smallness, as follows from e. (57). On the other hand, ths non-perfecton leads to the errors n the end-effector poston and orentaton wth the frst order of smallness. For ths reason, the geometrcal errors from manufacturng and assemblng do not appear drectly n the fnal expresson for the stffness matrx (5). Nevertheless, the above mentoned devatons effect the eulbrum confguraton wth respect to whch the Jacobans and the stffness coeffcents are computed. For ths reason, the authors prefer to take nto account such devatons of the second order smallness n order to detect some non-lnear phenomenon n the manpulator behavour, such as geometrcal bucklng studed n our prevous paper [1]. Hence, here the stffness model assemblng technue from Secton 4.1 has been extended for the case of parallel manpulators wth geometrcal errors n seral chans. In addton to computng of aggregated artesan stffness matrx, t allows us to evaluate nternal deflectons and forces/torues n jonts, as well as deflectons of the reference pont caused by geometrcal errors n knematc chans. In spte of ths ssue has essental practcal sgnfcance for evaluatng the desred tolerances n lnks/jonts geometry and correspondng nternal stresses n over-constraned mechansms, t has never been studed before. 2 (58)

22 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 4.3 Stffness model of non-perfect parallel manpulator under external loadng Let us focus on the aggregaton of stffness models of separate seral chans nto the stffness model of the whole parallel manpulator n the loaded mode. o solve ths problem, t s necessary to obtan the non-lnear force-deflecton relaton, whch takes nto account elastostatc propertes of all knematc chans, and to compute correspondng artesan stffness matrx. Let us assume that the end-ponts of all knematc chans are algned and matched n the same target pont t, whch corresponds to the desred end-platform locaton. hs pont s assumed to be known and allows us to compute, from the nverse knematc model, the actuator and passve jont coordnates defnng nomnal confguratons of the chans (, ). It s also assumed that the stffness models of all knematc chans have been already obtaned usng technues proposed n Sectons 4.2 and are presented n the form of partal non-lnear force-deflecton relatons F f ( t t ) correspondng to the target pont t. It s evdent that the external loadng F changes the end-platform locaton t, hence t s reasonable to consder the set of locatons t n the neghborhood of target one. Under the above assumptons, for any gven pont t from neghborhood of t t s possble to compute both the partal forces F and correspondng eulbrum confguratons (, ). hen, n accordance wth the superposton prncple, the desred non-lnear force-deflecton relaton for the whole parallel manpulator can be found by straghtforward summaton of all partal forces F,.e. m F f t t (66) 1 where F denotes the total external loadng appled to the end-platform. As a result, correspondng curves can be obtaned by multple repetton of the above descrbed procedures for dfferent values of the end-platform locaton t. ( ) Furthermore, for each gven t, the stffness matrces K of all knematc chans can be computed usng expresson (5). hs allows us to compute the artesan stffness matrx K of the whole parallel manpulator as a sum m 1 ( ) K K (67) ( ) ( ) However, the matrces K and K defnng the "senstvty" of the chan jont coordnates (, ) to the end-platform dsplacement cannot be aggregated n ths way, they should be used separately to evaluate stresses n jonts/lnks and resstance of the chan confguratons wth respect to external loadng F ( ) ( ) ( ) ( ) ( ) τ J K t t ; K t t ; K t t (68) ( ) where J s Jacoban matrx of -th knematc chan wth respect to vrtual jont coordnates. It s worth mentonng that above t was mplctly assumed that the manpulator assemblng s euvalent to the algnng and matchng of the chan end-frames. o deal wth more general case, when the chans are connected to the dfferent ponts of the platform, t s necessary to slghtly modfy the chan geometrcal models and to re-compute the forces/torues and the stffness matrces by addng a vrtual rgd lnk connectng the end-pont of the chan and the reference pont of the platform (see Fgure 5 where these extensons are defned by the vectors v ). After the relevant transformatons, the above presented technue can be appled straghtforwardly. Besdes, here there are no evdent dfferences n stffness models aggregaton of perfect and non-perfect knematc chans. However, here the chan geometrcal errors are mplctly ncluded n the functons g (, ). In partcular for non-perfect chans, t s assumed that the nomnal values of the jont coordnates (, ) produce the end-pont locaton vector whch dffers from t : g (, ) t (69) where accumulates nfluences of all geometrcal errors on the end-pont locaton of -th chan. As a result, the end-platform cannot be located n the target pont t wthout external loadng,.e. 21 m 1 f t t (7) t t Moreover, wthout external loadng, the end-platform locaton t s dfferent from the target one t. he vector t can be computed from the euaton m f t t (71) 1

23 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 22 Fgure 7 Aggregaton of seral chans stffness models technue whch wll be consdered n the next sub-secton. orrespondng nternal forces F defnng the chan loadngs due to the geometrcal errors n the chans can be computed by smple substtuton t to the partal force deflecton relatons F f ( t t ) (72) t t It s obvous that the sum of the over-constraned. F s eual to zero but they produce stresses n the lnks and jonts f the parallel manpulator s Hence, the developed aggregaton technue allows us to obtan the non-lnear force-deflecton relaton for a parallel manpulator n the loaded mode as well as to compute artesan stffness matrces for any gven target pont t and gven set of the end-pont locatons {} t. hs technue s summarzed n Fgure Model nverson: complance propertes of parallel manpulator under the loadng he non-lnear force-deflecton relaton (66) allows us to evaluate the external force/torue F reured to locate the manpulator n the target pont t (assumng that the actuated coordnates are computed for the end-platform locaton t correspondng to the unloaded confguraton). However n practce, t s often necessary to determne the end platform resstance to the external loadng,.e. to compute the deflecton caused by the force F appled to the end-platform. he desred value can be found from the non-lnear complance model that n general case s expressed as 1 t f ( F t ) (73) and s defned by the nverse f 1 (...) whch for parallel manpulators usually exsts (due to over-constraned structure). In contrast, for seral chans wth passve jonts, the functon f 1 (...) cannot be computed snce the correspondng artesan ( ) stffness matrx K s sngular. It s obvous that n a general case, the functon f 1 (...) cannot be expressed analytcally. Hence, t s reured that a dedcated teratve procedure, whch s able to solve the non-lnear euaton (73) for t (assumng that F s gven). It s proposed here to apply a modfcaton of Newton-Raphson technue whch teratvely updates the desred value t n accordance wth the expresson f t t K t t F t t (74) 1 where t corresponds to the next teraton, K t t s the artesan stffness matrx computed n the pont t, and t denotes the unloaded locaton of the end-platform. For ths teratve scheme, t can be also used as the ntal value of t. Smlar to ( ) sub-secton 4.5.1, wthn each teratve loop, correspondng confguratons (, ), the loadngs F and stffness matrces K for each knematc chan are computed usng euatons (36), (5).

24 A. Klmchk, D. hablat, A. Pashkevch Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs 23 Fgure 8 Procedure for obtanng deflecton-force relaton for loaded parallel manpulator As t follows from the relevant study, convergence of ths teratve procedure s good enough f the functon f (...) s smooth and non-sngular n the neghborhood of t. If t s reured to mprove convergence, t s possble to apply the same technue as n Secton 3.3, when the force F s modfed from teraton to teraton n accordance wth the expresson F F, where a scalar varable s monotoncally ncreasng from up to 1. he stoppng crteron can be expressed n a straghtforward way as F f t t (75) F where s the desred accuracy. More detals presentaton of the developed teratve routnes s gven n Fgure 8 F Summarzng ths sub-secton, t s worth mentonng that the developed technue allows to obtan the desred non-lnear deflecton-force relaton descrbng the end-platform resstance wth respect to the external force for a gven t, whch corresponds to the unloaded (nomnal) manpulator confguraton. he above presented results can be used both for stffness modelng of manpulator under the loadng and for complance error compensaton. 5 Applcaton examples: parallel translatonal manpulator 5.1 Non-perfect parallel manpulator wthout external loadng Let us llustrate the developed stffness model aggregaton technue by examples that deal wth assemblng of Orthoglde parallel translatonal manpulator wth geometrcal errors n knematc chans (Fgure 9). he manpulator consst of three knematc chans wth one translatonal actuator along the axes x, y or z, two passve U-jonts (or two separate rotatonal jonts) and knematc parallelogram between them. It should be mentoned that two chans wth knematc parallelograms would be suffcent to restrct the moton n translaton. However n order to ncrease the manpulator stffness, the thrd actuated chan has been ntroduced that mpose redundant constrants on the moble platform,.e. the Orthoglde manpulator has an over-constraned structure. hs manpulator has the workng area of sze mm wth the transmsson factor from.5 to 2; the detaled geometrcal parameters are presented n [58], the lnk stffness matrces were computed usng an approach developed n [34], where ther numercal values are presented as well. Let us assume that the manpulators have geometrcal errors n the knematc chans, whch have effects on the end-pont locaton and provoke nternal loadngs n the jonts. Q 2 y z x y Q x Q 4 Q 3 z Q 1 (a) Photo (b) AD-model Fgure 9 Photo and AD models of Orthoglde manpulator