Determination of inertia tensor co-ordinates of manipulators on the basis of local matrices of homogeneous transformations

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1 Mechnics nd Mechnicl Engineering Vol. 10, No 1 (2006) c Technicl University of Lodz Determintion of inerti tensor co-ordintes of mniultors on the bsis of locl mtrices of homogeneous trnsformtions Przemys lw Szumiński, Tomsz Kitnik Division of Dynmics, Technicl University of Lodz Stefnowskiego 1/15, Lodz, Polnd e-mil: rzeszum@.lodz.l Received (20 Aril 2006) Revised (31 My 2006) Acceted (8 June 2006) The im of this er is to resent n lgorithm for the determintion of co-ordintes of inerti tensors of mniultor links nd objects mniulted, which emloys homogeneous trnsformtions. A division of mniultor link (n object mniulted) nd locl systems of reference, which corresond to this division, hve been introduced. An introduction of dditionl locl systems of reference llows for their most dvntgeous orienttion in order to clculte the co-ordintes of inerti tensors of rts nd elements, whose forms nd shes often cuse serious roblems during the clcultion of inerti tensors with the trditionl method. In order to trnsform geometricl dimensions of solid bodies nd moments of inerti, n liction of mtrices of homogenous trnsformtions, tyicl of the descrition of mniultor kinemtics, hs been suggested. Owing to the emloyment of homogenous trnsformtions nd their mings, the roosed lgorithm mkes the determintion of co-ordintes of inerti tensors much esier, esecilly in the cse their she is comlex. Keywords: Inerti tensor, homogeneous trnsformtions, mniultors 1. Introduction If rigid solid body tht rottes round fixed xis of rottion is considered, then the notion of moment of inerti is used. For rigid body tht erforms stil motion, the notion of tensor of inerti is introduced s sclr generlistion of the moment of inerti. Among fundmentl roerties of inerti tensors of solids, one cn mention their deendence on the osition nd orienttion of the system of reference with resect to which the tensor of inerti is defined. A clssicl wy of the determintion of inerti tensor rmeters of the solid body, in the cse of mniultors - usully of link or n object mniulted, consists in the clcultion of its co-ordintes with resect to the locl system of reference of the solid tht is fixed it the solid centre of grvity [1, 4, 6, 7, 11]. This rocedure results from the nottion of generlly cceted equtions of mniultor dynmics. In the

2 Determintion of inerti tensor develoed lgorithm for the determintion of inerti tensor rmeters, method tht introduces subdivision of the mniultor link (mniulted object) nd locl systems of reference is roosed. The link (mniulted object) is structurlly divided into elements, nd they in turn, re comosed of rts. This division is ech time determined by the design structure of the mniultor link or the mniulted object under considertion. This method enbles the determintion of the co-ordintes of inerti tensors of links (mniulted objects), the elements nd rts tht re secified in them, with resect to locl systems of reference tht re rbitrrily orientted in sce. These systems of reference hve their beginnings in the centres of grvity of resective elements nd rts of the link (object). In order to trnsform the geometricl dimensions of solids nd their moments of inerti, usge of mtrices of homogeneous trnsformtions is suggested [5, 2, 4, 10]. The roosed lgorithm simlifies significntly the determintion of the co-ordintes of inerti tensors of the link nd the mniulted object, esecilly when their she is comlex. Mtrices of homogeneous trnsformtions, well known in the mniultor kinemtics, re emloyed here. An exemlry nlysis of the rmeters of inerti tensors of links nd mniulted object is resented. 2. Structurl division of mniultor links. Locl systems of reference of links Let us consider n rbitrry link i of the mniultor, which we subdivide into j i elements, Fig. 1. Forms nd number of elements deend on the design of the link under considertion. In turn, ech element hs individul rts, which re for instnce housings of berings, bering journls, drive trnsfer elements or griing device elements. We ssume tht the volumes of the rts of the bering systems nd drive trnsfer systems connected with the link for both the kinemtic irs nd the link body, s well s resective mteril roerties, re known. Let us secify elements k, r, l from the totl number of elements j i of the link i, where denotes ny element of the link i, r refers to the element of the link i tht includes the locl system of reference of the link i, nd l stnds for the element of the link i tht includes the locl system of reference of the link i 1/i + 1. Further, let us ssume thus tht ech element of the link i hs in turn the secified rts q k, q r, q l, corresondingly. In generl, ny element of the link i hs secified rts q. The system of co-ordintes x r y r z r 0 r is the locl system of reference of the element r, which includes the locl system of reference x i y i z i 0 i t the sme time. In Fig. 1: L i - vector defining the distnce of centres of locl systems of reference of the link i nd i + 1/i 1 long the stright line tht connects them; α i, β i - design dt of the link (geometricl distnce of the oint 0 from the element of the link body); V i - sum of volumes of secified rts of the -th link element; c r, c k, c l - ositions of resultnts of centres of grvity of the elements r, k, l, resectively. These oints re resultnt ositions of centres of grvity of bering systems nd systems of drive trnsfer relted to the link; 0 r, 0 l - centres of locl systems of reference of the link i nd i + 1/i 1; S i - osition of the link centre of grvity. Let us introduce dditionl locl systems of reference for elements k nd l, resectively. The systems of co-ordintes x k y k z k 0 k nd x l y l z l 0 l re locl systems of reference of the k-th nd l-th element of the link i, corresondingly.

3 140 Szumiński P. nd Kitnik T. Figure 1 Structurl division of the i-th link of the mniultor The most dvntgeous wy of the selection of locl systems of link elements is such when one xis of ech system lies on the stright line tht connects centres of these systems. The locl system of the link body should be selected in the hlflength of L i (the stright line tht connects 0 r 0 l ). Furthermore, it is dvntgeous to ssume locl systems of reference of individul elements of the link in such wy tht they hve resective xes rllel nd senses consistent with ech other. The element tht includes the locl system of reference of the link hs lso locl system of reference with its beginning fixed in the centre of the locl system of reference of this link. A selection of locl systems of reference of individul rts nd elements of the link is determined by the most dvntgeous determintion of corresonding tensors of inerti (determintion of geometricl dimensions nd limits of integrtion). Emloying mtrix of rottion (17), we cn trnsform the geometricl dimensions nd the limits of integrtion (Section 6) between the locl systems of reference of rts nd elements nd the system of co-ordintes of the link i. The osition of the resultnt centre of grvity of the -th element hs been described by the vector: r i c = [rc, x rc, y rc] z T (1) in the locl system of reference of the -th element of the link i. The ositions of resultnt centres of grvity of the elements r, k, l re described by the following vectors: r i cr = [r x cr, r y cr, r z cr] T, r i ck = [r x ck, r y ck, rz ck] T, r i cl = [r x cl, r y cl, rz cl] T (2) in the corresonding locl systems of reference of these elements.

4 Determintion of inerti tensor Generlly, the osition of the centre of grvity of the link i with resect to the locl system of reference of the element tht includes the locl system of reference of the link i (tht is to sy, of the element r), cn be determined from the reltionshi: x r si = j i j i ρ i V i x i c ρ i V i, y r si = j i j i ρ i V i y i c ρ i V i, z r si = j i j i ρ i V i z i c ρ i V i (3) where: = 1, 2,..., r,..., k,..., l,..., j i, j i - number of elements of the model of the link i; ρ i - mteril density of the -th element of the link i; V i - volume of the -th element of the link i; [ x i c, y i c, z i c] T - vector of the osition of the resultnt centre of grvity of the -th element in the locl system of reference of the element r (tht includes the locl system of reference of the link i), nd: j i V i = j i q q m i,q ρ i q, (4) where: m i,q - mss of the q-th rt of the -th element of the link i; ρ i q, - mteril density of the q-th rt of the -th element of the link i. Assuming the nottions s in Fig. 1, we cn exress the volume of the link i s follows: j i V i = V i r + V i k + V i l = q r q=1 m ir,q ρ i q,r + q k q=1 m ik,q ρ i q,k q l + q=1 m il,q ρ i q,l (5) where: j i = 3; Vr i, Vl i - volume of the housing nd the bering system nd elements of the drive relted to the kinemtic ir i nd i + 1/i 1, resectively, Vk i - volume of the body of the link i. Choosing locl systems of reference of the link i s in Fig. 1 nd Fig. 5, for constnt vlue of the mteril density of the link, we cn exress equtions of co-ordintes (3) in the form: co-ordinte x r si : x i cr = r x cr, x i ck = L i 2 + rx ck, xi cl = L i + r x cl x r si = V i r rx cr 1 2 V i k Li+V i k rx ck +V i l rx cl V i l Li V i r +V i k +V i l (6) wheres: x r si = 1 2 L i <=> r x cr + r x ck + r x cl = 0 V i r = V i k = V i l (7) co-ordinte y r si : y i cr = r y cr, y i ck = ry ck, yi cl = ry cl y r si = V i r ry cr +V i k ry ck +V i l ry cl V i r +V i k +V i l (8)

5 142 Szumiński P. nd Kitnik T. wheres: y r si = 0 <=> V i r r y cr + V i k ry ck + V i l ry cl = 0 : rcr y = r y ck = ry cl = 0 ry cr + r y ck + ry cl = 0 V r i = Vk i = V l i rcr y = 0 V i k = ry Vl i cl r y ck r y ck = 0 V i r = ry Vl i cl rcr y (9) r y cl = 0 V i r V i k = ry ck r y cr co-ordinte z r si : wheres: z i cr = r z cr, z i ck = rz ck, zi cl = rz cl z r si = V i r rz cr +V i k rz ck +V i l rz cl V i r +V i k +V i l z r si = 0 <=> V i r r z cr + V i k rz ck + V i l rz cl = 0 : (10) rcr z = rck z = rz cl = 0 rz cr + rck z + rz cl = 0 V r i = Vk i = V l i rcr z = 0 V i k = rz Vl i cl rck z (11) rck z = 0 V i r = rz Vl i cl rcr z r z cl = 0 V i r V i k = rz ck r z cr The osition of the centre of grvity of the link i with resect to the locl system of reference of the element r cn be exressed s: r r si = xr si y r si z r si = V i r rx cr +V i k (rx ck L i 2 )+V i V i r +V i k +V i l V i r ry cr +V i k ry ck +V i l ry cl Vr i+v k i+v l i V i r rz cr +V i k rz ck +V i l rz cl Vr i+v k i+v l i l (rx cl L i) (12) In generl, the osition of the centre of grvity of the link i with resect to the locl system of reference of the link i (x i y i z i 0 i ), Fig. 2, cn be written s: r si = A o r r si (13)

6 Determintion of inerti tensor where: A o - mtrix of rottion of the co-ordinte system of the link, x i y i z i 0 i, with resect to the osition of the locl system of the element r. Knowing the descrition of n rbitrry vector with resect to the locl system of the element r, we intend to describe this vector in the system of reference of the link i, s the beginnings of both the systems coincide. This trnsformtion is ossible rovided the descrition of the orienttion of the locl system r with resect to the system of co-ordintes of the link i is known. This orienttion is defined by the mtrix of rottion A o, whose columns re versors of the system of reference of the element r described in the system of reference of the link, nd rows re versors of the system of the link r described in the system of the element r. Owing to the mtrix of rottion A o, descrition of the orienttion of the system of the element r with resect to the system of reference of the link i is ossible, i.e. trnsformtion of the orienttion of ny rbitrry vector described with resect to the locl system of reference of the element r to the locl system of reference of the link i. In order to describe the orienttion of the rottion in the three-dimensionl sce, one should ) b) Figure 2 Trnsformtions of locl systems of reference remember tht rottions in generl re not commuttive (multiliction of mtrices does not exhibit the roerty of commuttion). Therefore, in order to describe the orienttion of the locl system of reference of the element r with resect to the locl system of reference of the link i, the locl system of the reference of the link i should be rotted with resect to the resective xes of the locl system of the element r of this link. In order to trnsform (rotte) the locl system of reference of the link i to the locl system of the reference of the element r, the following rottions round

7 144 Szumiński P. nd Kitnik T. the corresonding xes of the locl system, Fig. 2b, hve been erformed in the following sequence: - rottion round the xis x r by n ngle ± γ - rottion round the xis y r by n ngle ± α - rottion round the xis z r by n ngle ± δ (14) where, deending on the ositions of systems of reference with resect to ech other, the ngles α, δ nd γ cn ssume ositive or negtive vlues. Angles consistent with the trigonometric direction of the rottion will be considered ositive, if we ssume the xis of rottion of the system directed t the erson looking t it. Ech of secified rottions is erformed long resective xes of the defined system of reference of the element r. The mtrix of rottion of the locl system of reference of the link i with resect to the locl system of reference of the element r cn be written in the generl form s follows: A o = A δ A α A γ (15) The mtrices of elementry rotting trnsformtions re s follows: A γ = cos(α) 0 sin(α) 0 cos(γ) sin(γ), A α = sin(γ) cos(γ) sin(α) 0 cos(α) A δ = cos(δ) sin(δ) 0 sin(δ) cos(δ) , (16) If we tret rottions s oertors [2], we will erform rottions A γ, A α nd A δ. After multilictions ccording to Eq. (15), we obtin: cos(α) cos(δ) A o = sin(δ) cos(α) sin(α) sin(α) sin(γ) cos(δ) sin(δ) cos(γ) sin(α) sin(δ) sin(γ) + cos(δ) cos(γ) sin(γ) cos(α) sin(α) cos(δ) cos(γ) + sin(δ) sin(γ) sin(α) sin(δ) cos(γ) sin(γ) cos(δ) cos(α) cos(γ) Mtrix (17) is correct only for rottions erformed in the sequence given in (14). If we emloy mtrix (17), then Eq. (13) tkes the form: cos(α) cos(δ) r si = sin(δ) cos(α) sin(α) sin(α) sin(γ) cos(δ) sin(δ) cos(γ) sin(α) sin(δ) sin(γ) + cos(δ) cos(γ) sin(γ) cos(α) sin(α) cos(δ) cos(γ) + sin(δ) sin(γ) sin(α) sin(δ) cos(γ) sin(γ) cos(δ) cos(α) cos(γ) xr si y r si z r si (17) (18)

8 Determintion of inerti tensor Assuming the equlity α = δ = γ = 0 the mtrix of rottion hs the following form: A 0 = (19) then r si = r r si (20) This is the cse when the locl system of reference of the link is the locl system of reference of the link element within which there is the locl system of reference of this link. 3. Algorithm for the determintion of rmeters of inerti tensors of mniultor links As we know, determintion of the tensor of inerti of the link i consists in determintion of co-ordintes of the tensor of inerti in the system of co-ordintes, whose beginning lies in the centre of mss of the link nd whose xes hve consistent senses nd re rllel to the corresonding xes of the locl system of co-ordintes relted to this link. The generl form of the inerti tensor of the link i is s follows [5, 8, 9]: I si = B x si B xysi B xzsi B xysi B ysi B yzsi B xzsi B yzsi B zsi (21) where: B αsi - mss moment of inerti of the i-th link with resect to the lnes erendiculr to the xis α, α = {x, y, z}; B αβsi - mss centrifugl moment of inerti of the i-th link with resect to the lnes erendiculr to the xis α nd β, α = {x, y, z} nd β = {x, y, z} nd α β. The system of reference (xyz) si hs its beginning in the centre of mss of the link i nd xes rllel nd with consistent senses to the resective xes of the system relted to the link (tht is to sy, rllel to the xis of the system x i y i z i 0 i ). Ech link s mteril body cn be divided in mind into indefinitely smll elements. Let us denote the mss of one of such elements of this body by dm nd let us ssume tht it hs the mteril density ρ nd the volume dv, then dm = ρdv. The tensor of inerti of ny q-th rt of the -th element of the link i ssumes the form: y Iq, i = ρ i m 2 + zm 2 x m y m x m z m q, y m x m x 2 m + zm 2 y m z m dv (22) Vq, i z m x m z m y m x 2 m + ym 2 where: m - ny oint of the link i; ρ i q, - mteril density of the q-th rt of the -th element of the link i; Vq, i - volume of the q-th rt of the -th element of the link i, dv = dxdydz - elementry volume; m i, V i - mss nd volume of the i-th link, resectively. Let us denote the vector of the osition of n rbitrry oint m of the link in the system of co-ordintes whose beginning is in the centre of grvity of the rt q of the element nd whose xes hve consistent sensors nd re rllel

9 146 Szumiński P. nd Kitnik T. to the corresonding xes of the locl system of reference of the link i (system of co-ordintes of the link i) by r m = [x m, y m, z m ] T. The vector r m cn be written if we know the osition of the oint m in the locl system of reference of the q-th rt of the -th element. The reltionshi between vectors r m nd r ml hs been defined s: r m = A o r ml (23) where: A o mtrix of rottion of the locl system of reference of the link i, trnsformed to the centre of grvity of the considered (secified) element or rt of the link, to the locl system of reference of this element or rt; r ml - vector of the osition of the oint m of the link in the locl system of co-ordintes of the rt q of the -th element, ( r ml = [x ml, y ml, z ml ] T ). Substituting reltionshi (17) into Eq. (23) we obtin: x m y m z m = r m = A o r ml cos(α) cos(δ)x ml + [sin(α) sin(γ) cos(δ) sin(δ) cos(γ)]y ml sin(δ) cos(α)x ml + [sin(α) sin(δ) sin(γ) + cos(δ) cos(γ)]y ml sin(α)x ml + sin(γ) cos(α)y ml + cos(α) cos(γ)z ml +[sin(α) cos(δ) cos(γ) + sin(δ) sin(γ)]z ml +[sin(α) sin(δ) cos(γ) sin(γ) cos(δ)]z ml (24) Hving substituted (24) into (22), for the cse when α = 0 nd remembering to trnsform the limits of integrtion to the locl system of reference of the rt in order to simlify the clcultions, we obtin s result of these trnsformtions wht follows: I i q, = ρ i q, V i q, sym. b c d e f dv (25) where:

10 Determintion of inerti tensor = sin(δ)[sin(δ)x ml + 2 cos(δ)(cos(γ)y ml sin(γ)z ml )]x ml + cos 2 (δ)[cos(γ)y ml sin(γ)z ml ] 2 + [sin(γ)y ml + cos(γ)z ml ] 2 b = sin(δ) cos(δ)[x 2 ml (sin(γ)z ml cos(γ)y ml ) 2 ]+ x ml (sin 2 (δ) cos 2 (δ))[sin(γ)z ml cos(γ)y ml ] c = cos(δ)[cos(δ)x ml + 2 sin(δ)(sin(γ)z ml cos(γ)y ml )]x ml + sin 2 (δ)[cos(γ)y ml sin(γ)z ml ] 2 + [sin(γ)y ml + cos(γ)z ml ] 2 d = sin(δ)[sin(γ) cos(γ)(z 2 ml y2 ml ) + y mlz ml (sin 2 (γ) cos 2 (γ))]+ cos(δ)[sin(γ)y ml + cos(γ)z ml ]x ml (26) e = cos(δ)[sin(γ) cos(γ)(y 2 ml z2 ml ) + y mlz ml (cos 2 (γ) sin 2 (γ))]+ sin(δ)[sin(γ)y ml + cos(γ)z ml ]x ml f = [cos(δ)x ml sin(δ cos(γ)y ml + sin(δ sin(γ)z ml ] 2 + [sin(δ)x ml + cos(δ) cos(γ)y ml cos(δ) sin(γ)z ml ] 2 In turn, the mtrix Iq,,s i is clculted with resect to the centre of grvity of the link i, wheres the mtrix Iq,s i - with resect to the centre of grvity of the element of the link i, using the Huygens-Steiner theorem for this im [3]. Next, the tensor of inerti of the link i (I is ) is determined s sum of the mtrix Iq,,s i (summtion of the corresonding terms of the mtrix) of the subsequent rts of the resective elements of the link i (or the mtrix I,s i of the subsequent elements of the link i) nd the tensor of inerti of the element (Is) i s sum of the mtrix Iq,s i of the subsequent rts of the -th element with resect to the centre of grvity of the element. The determintion of tensors of inerti of -th elements of the link i llows for n nlysis of the fctors tht ffect the reltions of inerti tensors of individul elements of the link. An exemlry nlysis, bsed on the resented lgorithm, is crried out in Section Algorithm for the determintion of rmeters of inerti tensors of mniulted objects Let us consider the mniulted object shown in Fig. 3. Let us ssume tht the osition of the centre of grvity of the mniulted object is dislced with resect to the beginning of the locl system of reference of the lst link of the kinemtic mniultor chin in the form of the vector r. We wnt to describe the tensor of inerti of the mniulted object with resect to the system of reference with the beginning in the centre of grvity of this object nd with xes tht hve consistent senses nd re rllel to the corresonding xes of the locl system of reference of the lst link of the mniultor kinemtic chin (link n). It llows for the determintion of the kinemtic energy of the mniulted object, emloying the knowledge of kinemtics of the lst link of the mniultor kinemtic chin thus. In order to simlify the clcultions of the co-ordintes of the inerti tensor of the mniulted object, it is dvntgeous to introduce locl system of reference of the object with the beginning in the centre of grvity of the mniulted object

11 148 Szumiński P. nd Kitnik T. Figure 3 Locl systems of reference relted to the object mniulted (x y z 0 ). The orienttion of the locl system of reference of the mniulted object should be selected in the most dvntgeous wy from the viewoint of the clcultions of the inerti tensor of the mniulted object with resect to its own system of reference. Let us denote the vector of the osition of the centre of grvity of the mniulted object in the locl system of reference of the n-th link of the mniultor kinemtic chin, Fig. 3, by r = [x s, y s, z s ] T. The tensor of inerti of the mniulted object with resect of the object centre of grvity, in its locl system of reference (x y z 0 ), ssumes the following form in generl: I l = Bx l Bxy l Bxz l Byx l By l Byz l Bzx l Bzy l Bz l = m i 2 x Bxy l Bxz l Byx l m i 2 y Byz l Bzx l Bzy l m i 2 z (27) where: B α, B αβ, α = {x, y, z}, β = {x, y, z}, α β - mss moments of inerti of the object mniulted with resect to the locl system of reference of the object locted in its centre of grvity; they deend on the she of the object; m - mss of the object mniulted; i 2 α - resective rdii of inerti. Then, fter trnsformtions (17) of geometricl dimensions nd limits of integrtion (Section 6), we cn trnsform the quntities written in the locl system of reference of the object to the locl system of reference locted in the centre of grvity of the mniulted object nd with xes tht hve consistent senses nd re rllel to the resective xes of the locl system of co-ordintes of the lst link of the mniultor kinemtic chin (see Eq. (25) s n exmle). As result, we obtin the tensor of inerti of the mniulted object I with resect to the system of reference fixed in the centre of grvity of the object nd with xes of consistent

12 Determintion of inerti tensor senses nd rllel to the corresonding xes of the lst link of the kinemtic chin: I = Bx Bxy Bxz Byx By Byz Bzx Bzy Bz (28) ) b) c) Figure 4 The mniultor griing device - mniulted object system. Exemlry selection of locl systems of reference for the cse of the object mniulted in the form of homogeneous cube: ) mtrix of trnsformtion of locl systems of reference ccording to Eq. (19); b), c) mtrix of trnsformtion of locl systems of reference ccording to Eq. (30). The locl system of reference of the mniulted object (x y z 0 ) is its min centrl system of inerti In the cse when the she of the object mniulted is such tht selection of the locl system of reference - the most dvntgeous from the viewoint of the determintion of its mss moments of inerti - is such tht the corresonding xes of the locl system of reference of the object nd the locl system of reference of the lst link of the mniultor kinemtic chin re rllel nd hve consistent senses, mtrix (17) of trnsformtions of these systems ssumes form (19). Then,

13 150 Szumiński P. nd Kitnik T. the equlity of tensors of inerti tkes lce: I = I l (29) Let us consider homogeneous hexgon whose side length is w s the mniulted object. Assuming the locl systems of reference s in Figs. 4 nd 4b,c, mtrices A o ssume the forms (19) nd (30), resectively. Considering the cse resented in Figs. 4b,c, we erform rottion by n ngle α round the xis y of the locl system of reference of the lst link of the kinemtic chin with resect to the osition of the locl system of reference of the object mniulted. Emloying reltionshis (14) nd (15), the mtrix of rottion ssumes then the form: cos(α) 0 sin(α) A o = A α = (30) sin(α) 0 cos(α) In the cse under considertion, we will write Eq. (23) s follows: r n = A o r (31) After trnsformtions of Eq. (31), the comonents of the vector r ssume the following forms, corresondingly: x n = x cos(α) + z sin(α) y n = y z n = z cos(α) x sin(α) As n exmle, we will trnsform the geometricl dimensions of the osition of some selected, exemlry oints of the object under mniultion in the locl system of reference of the lst link of the mniultor chin: oint 1 : x =, z = 0 => x n = x cos(α), z n = x sin(α) (32) oint 2 : x = 0, z = b => x n = z sin(α), z n = z cos(α) (33) oint 3 : x =, z = b => x n = cos(α) + b sin(α), z n = b cos(α) sin(α) In the cse of the object mniulted under considertion = b = w /2. In turn, the tensor of inerti of the mniulted object with resect to the locl system of reference lced in the centre of grvity of the object nd with xes tht hve consistent senses nd re rllel to the resective xes of the locl system of the lst link of the mniultor kinemtic chin, for the cse resented in Fig. 4, ssumes the following form: where B x = B y = B z = mw2 6. I = I l = B x B y B z (34)

14 Determintion of inerti tensor Exmle of the nlysis of mniultor inerti tensors. Criteri for the selection of ositions of centres of grvity of mniultor links Let us focus on n exemlry nlysis of tensors of inerti. Let us tke n industril mniultor NM7MAR, whose kinemtic schemtic view is resented in Fig. 5, due to the simlicity of its design nd clerness of the nlysis. Figure 5 Kinemtic scheme of the NM7MAR mniultor. Selection of systems of co-ordintes of links Figure 6 She of the -th element of the i-th link of the exemlry mniultor Considering the structure of links of this mniultor, three elements in the form of homogeneous cubicoids (tking into ccount rts tht form bering systems nd their housings nd systems of drive trnsfer) hve been secified in ech link. Let us ssume the generl cse of the -th element of the link i. Let us ssume the geometricl nottions of the -th element s in Fig. 6. Thus, hving in mind the ssumtions concerning links of the mniultor under nlysis, we hve i = 1, 2, 3; = 1(r), 2(k), 3(l) - -th element of the link, m i - mss of the -th element of the

15 152 Szumiński P. nd Kitnik T. link, nd nlysing Figs. 1, 6: V i m i = ρ i i b i h i for i = 1, 2 nd = k => i = L i α i β i for i = 3 nd = k => h i = L i α i β i (35) Emloying reltionshi (22), the tensor of inerti of the -th element with resect to the locl system of co-ordintes of the link i tht is locted in the centre of grvity of the element, Fig. 6, ssumes the form in generl: Is i = (y2 m + zm)dm 2 x m y m dm x m z m dm x m y m dm (x 2 m + zm)dm 2 y m z m dm (36) x m z m dm y m z m dm (x 2 m + ym)dm 2 In the mniultor under considertion, owing to the ssumed ositions of locl systems of reference, the following equlity holds (from Eq. (23)): r m = r ml (37) Assuming tht the vector r i c = [r x c, r y c, r z c] T defines the osition of the centre of grvity C of the element in its locl system of reference, then the tensor of inerti I i s with resect to the locl system of reference x y z 0 ssumes the form: 2 Is i = 4 m i 12 (b2 i + h 2 i) + m i `(r y c ) 2 + (r z c) 2 m ir x cr y c m i r x cr z c m i r x cr z c m ir y cr z c m i rcr x y 12 (2 i + h 2 x i) + m i `(rc ) 2 + (rc) 2 z m 3 i rcr y c z m i m i 12 (2 i + b 2 i) + m i `(r x c ) 2 + (r y c) 2 where: i = 1, 2, 3; = r, k, l; m i = ρ i i b i h i wheres for: i = 1, 2, = k => i = L i α i β i, m i = ρ i b i h i (L i α i β i ); i = 3, = k => h i = L i α i β i, m i = ρ i i b i (L i α i β i ). The tensor of inerti I i s of the -th element of the link i is equl to: Is i = 5 (38) m i 12 (b2 i + h2 i ) 0 0 m 0 i 12 (2 i + h2 i ) 0 (39) m 0 0 i 12 (2 i + b2 i ) The determintion of the tensor I i,s consists in the determintion of the tensor of inerti of the element (in the mniultor under nlysis = 1(r), 2(k), 3(l)) with resect to the centre of grvity of the link i. The vectors of loctions of centres of grvity of the -th element of the link i hve the form: r i c = [r x c, r y c, r z c] T (40) where: i = 1, 2, 3; = r, k, l; in the locl systems of reference of the resective elements of the link. The vectors of ositions of the locl systems of reference of individul elements of

16 Determintion of inerti tensor the link in the locl system of reference of elements (link) tht is reduced to the centre of grvity of the link tke the form: r i o = [x i o, y i o, z i o] T (41) in the cse of the nlysed form of the link i: r i or = [x i or, y i or, z i or] T, r i ok = [x i ok, y i ok, z i ok] T, r i ol = [x i ol, y i ol, z i ol] T (42) The tensor of inerti I i,s is equl to: I,s i = m i 12 (b2 i + h2 i ) + m i[(r y c + y i o) 2 + (r z c + z i o) 2 ] m i [(r x c + x i o)(r y c + y i o)] m i [(r x c + x i o)(r z c + z i o)] m i [(r x c + x i o)(r y c + y i o)] m i 12 (2 i + h2 i ) + m i[(r x c + x i o) 2 + (r z c + z i o) 2 ] m i [(r y c + y i o)(r z c + z i o)] (43) m i [(r x c + x i o)t(r z c + z i o)] m i [(r y c + y i o)(r z c + z i o)] m i 12 (2 i + b2 i ) + m i[(r x c + x i o) 2 + (r y c + y i o) 2 ] The tensor of inerti of the i-th link is thus equl to: I si = I i,s (44) where: = r, k, l. Emloying reltionshi (43), Eq. (44) ssumes the form: I si = Bx i Bxy i Bxz i Bxy i By i Byz i Bxz i Byz i Bz i (45) where: i = 1, 2, 3; = r, k, l; B i x, B i y, B i z - mss moments of inerti of the link i with resect to the xis x i, y i, z i, resectively; B i xy, B i yz, B i xz - centrifugl moments of inerti of the link i with resect to the lnes of the system of co-ordintes, whose beginning lies in the centre of grvity of the link,

17 154 Szumiński P. nd Kitnik T. wheres: Bx i = m i [ 1 12 (b2 i + h2 i ) + (ry c + yo) i 2 + (rc z + zo) i 2 ] By i = m i [ 1 12 (2 i + h2 i ) + (rx c + x i o) 2 + (rc z + zo) i 2 ] B i z = m i [ 1 12 (2 i + b2 i ) + (rx c + x i o) 2 + (r y c + y i o) 2 ] B i xy = m i (r x c + x i o)(r y c + y i o) (46) B i xz = B i yz = m i (r x c + x i o)(r z c + z i o) m i (r y c + y i o)(r z c + z i o) Let us consider n exemlry link of the mniultor under nlysis, Fig. 7. Figure 7 Design scheme of the i-th link of the mniultor under nlysis. Systems of co-ordintes corresond to links 1 nd 2 For the links of the mniultor under nlysis, let us ssume the following volumetric reltionshis of the corresonding elements of the link: V r = V k = V l (47) nd of the comonents of the vectors of ositions of centres of grvity of individul

18 Determintion of inerti tensor link elements in their locl systems of reference, resectively: r x cr + r x ck + rx cl = 0 r y cr + r y ck + ry cl = 0 r z cr + r z ck + rz cl = 0 (48) As result of the ssumtions mde (Eqs. (47), (48) nd (6)-(12)), the osition of the centre of grvity of the i-th link of the mniultor in the oint whose coordintes re s follows: i = 1, 2 => ( Li 2, 0, 0) i = 3 => (0, 0, L i 2 ) (49) hs been determined in corresonding, locl systems of reference of links. 6. Trnsformtions of dimensions between locl systems of reference A trnsformtion of limits of integrtion will be resented on the exmle of the descrition of the orienttion of the vector tht is described by the limits of integrtion defined in the systems x ml y ml z ml 0 ml nd x m y m z m 0 m. Let us consider the cse shown in Fig. 8. Then, mtrix of rottion (17) ssumes the form: A o = cos(α) 0 sin(α) sin(α) 0 cos(α) After the substitution of Eq. (50) into reltionshi (23), we obtin: (50) x m = x ml cos(α) + z ml sin(α) y m = y ml =< w 2, w 2 > z m = x ml sin(α) + z ml cos(α) (51) Figure 8 Trnsformtions of dimensions between locl systems of reference

19 156 Szumiński P. nd Kitnik T. As result of the nlysis of Fig. 8, the co-ordintes of selected oints (i = 1, 2, 3, 4) of the solid body, in the locl system of reference x ml y ml z ml 0 ml, re resented in Tble 1. Tble 1 Considered oint x ml y ml z ml (i =1, 2, 3, 4) 1 w /2 0 w /2 2 w /2 0 w /2 3 w /2 0 w /2 4 w /2 0 w /2 After the trnsformtion of the locl systems of reference, ccording to Eq. (51), the co-ordintes of the oints under nlysis ssume the vlues included in Tble 2, resectively. Tble 2 Point x m y m z m (i = 1, 2, 3, 4) 1 w /2 sin(α) + w /2 cos(α) 0 w /2 sin(α) + w /2 cos(α) 2 w /2 sin(α) w /2 cos(α) 0 w /2 sin(α) + w /2 cos(α) 3 w /2 sin(α) + w /2 cos(α) 0 w /2 sin(α) w /2 cos(α) 4 w /2 sin(α) w /2 cos(α) 0 w /2 sin(α) w /2 cos(α) The equtions of limits of integrtion of the cube under considertion, generting the equtions of stright lines, with resect to the co-ordinte xes x nd z, in the form of system of equtions, hve the following form: [ xm = ± w 2 cos(α) + z mtg(α) z m = ± w 2 cos(α) x (52) mtg(α) for α (π/2, 3π/2). In the cse when the ngle α = π/4, Fig. 8, the co-ordintes of the oints under nlysis ssume vlues included in Tble 3. If we simlify eqution (52), the corresonding equtions of limits of [ x m = ± 2 2 w + z m z m = ± 2 2 w (53) x m integrtion hve the form: Let us consider homogeneous cube with side length w = nd let us ssume the locl systems of reference s in Fig. 9.

20 Determintion of inerti tensor Tble 3 Considered oint x m y m z m (i = 1, 2, 3, 4) 1 2/2w /2w /2w 4 2/2w 0 0 Figure 9 Scheme of trnsformtions of geometricl dimensions in locl systems of reference Solving the system of equtions tht corresonds to Eq. (52) (tht is to sy, we hve to remember bout the nottions of the xis of the ssumed orienttion of systems of reference), for instnce through substitution of Eq. (52b) into (52)), we obtin the co-ordintes of oint 1, Fig. 9, in the form: (x, y ) = ( 2 [sin(α) + cos(α)], 2 [cos(α) sin(α)] ) (54) nd other quntities shown in Figs. 9 nd 10 s well. For exmle, the moment of inerti of the secified rt of the solid body, whose cross-section in the lne x y is htched in the figures (for the cses from Figs.

21 158 Szumiński P. nd Kitnik T. 9, 9b, 9c nd 10) with resect to the xis z is equl to: B z + = ρ cos(α) 2 cos(α) 2 sin(α) 2 cos(α) 2 sin(α) [ 0 2 sin(α) y cot(α) 0 2 cos(α) +y tn(α) 0 (x 2 + y 2 )dx dy + (x 2 + y 2 )dx dy ]dz = ρ 5 24 (55) nd (Fig. 10): Figure 10 Scheme of trnsformtions of geometricl dimensions in locl systems B z = ρ sin(α) 2 cos(α) +y tn(α) 2 [cos(α) sin(α)] 2 cos(α) +y tn(α) 2 cos(α) 0 2 [cos(α) sin(α)] 2 sin(α) y tn 1 (α) tn 1 (α)y (x 2 + y 2 )dx dy + (x 2 + y 2 )dx dy + tn(α)y tn(α)y (x 2 + y 2 )dx dy dz = ρ5 24 (56) where: α (π/2, 3π/2), ρg mteril density of the cube. As result of the trnsformtion of geometricl dimensions, crried out on the bsis of Eqs. (23), (50) for the cses s in Figs. 9 nd 10, we obtin: B z = ρ (x 2 + y 2 )dxdydz = ρ 5 24 (57)

22 Determintion of inerti tensor Conclusions An nlyticl wy of the determintion of inerti tensor co-ordintes of mniultor links nd mniulted objects on the bsis of homogenous trnsformtion mtrices hs been resented. The min ide of this method consists in division of the solid body under nlysis into elements nd rts convenient for nlysis, n ttribution of locl systems of reference to them in wy tht mkes clcultions of element tensors of inerti esier, trnsformtions of the locl systems of reference, emloying mtrices of homogeneous trnsformtions, to the system of reference fixed in the centre of grvity of the link. Thus, using mtrices of homogeneous trnsformtions tht re commonly used in mniultor kinemtics, we cn simlify the lgorithm for the determintion of co-ordintes of tensors of inerti of links nd objects mniulted. Acknowledgments The investigtions were suorted by the Stte Committee for Scientific Reserch (KBN), Polnd, within Reserch Project 5T07A References [1] Armstrong B., Khtib O., Burdick J.: The exlicit dynmic model nd inerti rmeters of the PUMA 560 rm. Proc. of the IEEE, Int. Conference on Robotics nd Automtion, Sn Frncisco, (1986), [2] Crig J.J.: Introduction to robotics, Mechnics nd control, WNT, Wrsw (1993), Polnd. [3] Leyko J.: Dynmics of mteril systems, PWN, Wrsw, Polnd. [4] Morecki A., Knczyk J.: Fundmentls of robotics, Theory nd elements of mniultors nd robots, WNT, Wrsw (1999), Polnd. [5] Pul R.P.: Robot mniultor. Mthemtics, Progrmming nd Control, MIT Press, Cmbridge M.A., (1981). [6] Scivicco L., Sicilino B.: Modeling nd control of robot mniultors. New York, McGrw-Hill Com., (1996). [7] Song M.W., Vidysgr M.: Robot dynmics nd control. New York, J. Wiley, (1989). [8] Szwedowicz J.: Anlysis of dynmics of the mniultor with flexible links. Ph.D. Disserttion, Gdnsk Technicl University, (1991), Polnd. [9] Dwson D.M., Bridges M.M, Qu Z.: Nonliner control of robotic systems for environmentl wste nd restortion, Prentice Hll Inc., (1995). [10] Vukobrtovi M.: Dynmics of mniultion robots - theory nd liction, (1992). [11] Stiquito, Conrd J. M., Mills J. W.: Advnced exeriments with simle nd inexensive robot, Los Almitos, IEEE Com. Society, (1998).