Discrete Actuator Array Vectoreld Design for Distributed Manipulation Jonathan E. Luntz, William Messner, and Howie Choset Department of Mechanical En

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1 Discrete Actuatr Array Vectreld Design fr Distributed Manipulatin Jnathan E. Luntz, William Messner, and Hwie Chset Department f Mechanical Engineering Carnegie Melln University Pittsburgh, PA 11 jl9f@cmu.edu, bmessner@cmu.edu, chset@cs.cmu.edu Abstract he Mdular Distributed Manipulatr System (MDMS) is a macrscpic actuatr array which can manipulate bjects in the plane. he piecewisecnstant dynamics f manipulatin n the MDMS are develped based n an exact discrete representatin f the system. he resulting dynamics are inverted enabling the calculatin f an pen-lp vectr eld which prvides arbitrary unifrm bject dynamics. he vectr eld psitins, and under certain assumptins, rients bjects. 1 Intrductin he Mdular Distributed Manipulatr System (MDMS) is a macrscpic actuatr array which transfers, as well as manipulates, bjects in the plane, enhancing applicatins such as exible manufacturing and package handling systems. his system has been described in detail in previus wrk [, ] 1. Essentially, the MDMS cmprises an xed array f actuatrs (cells) each f which is an rthgnally munted pair r rller wheels whse cmbined mtin prvides a directable tractin frce t an bject resting n tp. In this system, several cells supprt a single bject that can be made t translate and rtate in the plane. Since sensing and actuatin are distributed, each f many bjects can be manipulated independently. (Figure 1). An 18 cell prttype is currently in peratin. In this paper, we derive the dynamics f mtin f an bject n a tw-dimensinal array f cells. he macrscpic scale f the system requires us t explicitly mdel the distributin f weight amng the discrete set f supprts as well as the tractin frces. 1 he MDMS is frmerly the Virtual Vehicle Figure 1: he MDMS: Several bjects can be translated and rtated, independently. Sectin describes sme f the prir wrk by thers and relates it t this wrk and the prir wrk f the authrs. he dynamic equatins derived in Sectin determine the mtin f an bject given a set f wheel speeds f each cell. In Sectin, we then slve the inverse prblem: determine the necessary wheel speeds t eect desired mtin. We design an pen-lp wheel velcity eld which brings an bject t a particular psitin by inverting the dynamic equatins. We shw that under certain assumptins this eld als brings an bject t a particular rientatin. Cncluding remarks are made in Sectin. Prir Wrk Bhringer, Dnald, et. al. [1] applied sensrless manipulatin ideas f Gldberg [] t an array f micrmechanical actuatrs which were used t psitin an rient very small bjects t ne f a nite number f rientatins. Kavraki [] supplied further analysis f micractuated systems using elliptical ptential elds t rient t a single rientatin. Due t the small scale f their applicatins, bth Bhringer and Kavraki made cntinuus eld assumptins in their

2 analysis. On the MDMS, hwever, a smaller number f cells supprt an bject, requiring explicit discrete mdeling f the system. In a previus paper by the authrs [] the rst step was taken in examining the dynamics f an bject carried by the MDMS, where the ne dimensinal mtin f the bject alng the array f cells was cnsidered. In that paper, the frces between each cell and the bject were calculated, and bth a culmb and a viscus-like frictin law were cnsidered. he resulting bject dynamics are that f a simple r damped harmnic scillatr, where the frequency, center f scillatin, and damping cnstant are parameters which change as the bject shifts frm ne set f supprts t anther. his simple scillatr behavir was als bserved in the prttype system. his wrk was extended int tw dimensins in a mre recent paper by the authrs []. ranslatinal frces and rtatinal trques are calculated as a functin f bject psitin. Similar mass-spring-damper behavir in the plane was bserved. his paper renes the authrs' previus wrk in tw dimensins by signicantly reducing the amunt f calculatin necessary t calculate frces and trques. A new result presented here is that this renement allws fr the inversin f the dynamics. A wheel velcity eld is thus generated which prduces desired bject dynamics with a single equilibrium psitin regardless f symmetry and cell reslutin, and, under certain assumptins, a single rientatin (within symmetry) t the reslutin f the cells. Dynamics f Manipulatin Initially, we will cnsider the dynamics f an array f cells transprting and rtating an bject in the plane while it rests n a single arbitrary set f cells. Fr this, the fllwing assumptins are made: Each rthgnally riented pair f wheels acts as a single supprt. Supprts act as springs t supprt the bject. he bttm f the bject is at. he speed f each wheel is cnstant. Hrizntal frce between each wheel and the bject is due t sliding frictin. Viscus frictin (prprtinal t speed) exists between the wheels and bject which is als prprtinal t the nrmal frce. he cmputatin f the hrizntal translatin and rtatin dynamics f the bject rst requires the use f the equilibrium f the bject and cnstitutive relatins fr the supprts. he hrizntal frces and trques are cmputed using a frictin law between the bject and wheels. his results in a net frce and trque acting n the bject as a functin f the bject's psitin. Ntatin: Nrmal math fnt represents scalar variables (e.g. s), arrwed nrmal math fnt represents vectrs (e.g. ~v), and bld fnt represents matrices (e.g. m). Subscripts x and y indicate x and y cmpnents, and subscript i indicates the ith cell. Fr example, V is a matrix made up f velcity (clumn) vectrs V ~ i fr each cell, with cmpnents V xi and V yi. V can als be said t be made up f cmpnent (rw) vectrs V ~ x and V ~ y listing the velcity cmpnents fr all the cells..1 Nrmal Frces Slving fr the n frces supprting the bject requires the cnsideratin f the equilibrium f the bject in bth the vertical (z) directin and in rtatin abut the x and y axes. Cnsider n cells arranged arbitrarily in the x-y plane having crdinates as entries f the matrix belw. X = ~x ~y = x1 ::: x n = X1 ~ ::: X y 1 ::: y ~ n n (1) An bject f weight W, whse center f mass is lcated at ~ X cm = x cm y cm resting n n f these cells, is supprted by vertical nrmal frces ~ N = N1 ::: N n. Vertical equilibrium f the bject requires that nx i=1 N i = W =[ 1 ::: 1 ] ~ N : () Rtatinal equilibrium abut the x and y axes requires that the mments induced by the nrmal frces abut any pint (in this case, the arbitrarily lcated rigin f ur crdinate system) sum t the mment abut this pint induced by the weight f the bject. herefre, nx nx i=1 i=1 N i y i = ~y ~ N = Wy cm ; and () N i x i = ~x ~ N = Wx cm : () At this pint in the develpment, there are n unknwns (the elements f ~ N), but nly three equatins

3 (,, and ) frm equilibrium. slve fr the remaining n, frces, exibility in the system must be cnsidered. Each supprt is assumed t be a spring, with Hke's Law (N i = K s z i ) representing the cmpressin f the ith cell under a nrmal lad. Physically, this exibility is either a exible suspensin under each wheel r, as in the prttype, exibility in the surface f the bttm f the bject. Assuming the bttm f the bject is nminally at, the exible cells cnfrm t the bttm f the bject t distribute the weight. All the supprting cells lie in this plane, which cnstrains the nrmal frces: N i + ax i + by i + c = () Equatins,, and alng with n instances f Equatin supply n+ equatins and n+ unknwns (n N i 's and plane parameters, a, b, and c). he matrix frm f this system f equatins is 1 ::: 1 x 1 y ::: 1 1 x n y n 1 ::: 1 x 1 ::: x n y 1 ::: y n A N 1. N n c b a ~N abc =. W Wx c Wy c ~W () A can be inverted t slve fr ~ N abc (which cntains ~N and a, b, and c.) Dene a matrix B. B = 1 ::: 1 x 1 ::: x n y 1 ::: y n () such that the expressin fr the matrix A is Inn B A = : (8) B he inverse f the matrix A exists if B has rank (which is true as lng as all the cells d nt lie n a line). he expressin fr the inverse is A,1 = "I, nn B BB,,1 B B BB #,1, BB,1 B,, BB,1 : : (9) Multiplying A by A,1 veries the result. Since ~ W nly multiplies nnzer elements int the right side f A,1, and nly the slpes (a, b, and c) result frm the lwer prtin f A,1, the calculatin N i V i f i X cm Figure : Interactin between wheel and bject. f ~ N uses nly the upper right partitin f A,1. ~N = B, BB,1 W Wy cm Wy cm, = W B 1. Planar Dynamics Xcm ~ A (1) he full planar dynamics invlve translatin and rtatin f the bject. he hrizntal frces are derived frm the nrmal frces thrugh the use f a viscustype frictin law (see Figure ). he hrizntal frce frm each cell fi ~ is prprtinal t a cecient f frictin, that cell's nrmal frce N i, and the vectr dierence between the velcity f the wheel and the velcity f the bject at the pint f the cell. his velcity dierence is a functin f bth the translatinal velcity f the bject ~ _ X cm, the velcity f the wheel V ~ i, the rtatin speed f the bject abut its center f mass!, and the psitin dierence between the cell and the center f mass X ~ i, X ~ cm. ~f i = ~V i, _ ~ Xcm +! 1,1 ~Xi, ~ X cm N i (11) he hrizntal frce frm each cell is summed up ver all the cells. Dene a wheel velcity matrix V as V = V1x V x ::: V nx : (1) V 1y V y ::: V ny Summing vectrially, the net hrizntal frce is ~f = V ~ N, _ ~ X cm W: (1) Observe that the net hrizntal frce is nt a functin f the bject's rtatin speed - the terms multiplying! are identically zer[]. Furthermre, the secnd term in this equatin is a dissipative linear damping term. he substitutin f ~ N frm Equatin 1

4 int Equatin 1 yields ~f = W VB, BB,1 1 1 k s, + W VB BB,1 1 ~f ~X cm, _ ~ Xcm W (1) where k s is a cnstantmatrix and f ~ is a cnstant 1 vectr. he matrix k s is essentially a matrix f spring cnstants, since it species frce as a linear functin f psitin. he vectr f ~ is an set frce. In tw dimensins, the trque each cell applies t the bject is the scalar crss prduct f the psitin vectr f the pint f applicatin f the frce, X ~ i, relative t the bject center f mass, X ~ cm, and the hrizntal frce vectr frm that pint, fi ~. After sme algebra[] the ttal trque n the bject is = R ~ N, ~ X ~ cm V N,! ~ ~X N,W ~ X ~ ~ cm X cm (1) where R i = X ~ i V ~ i denes R ~ and X i = X ~ i ~X i denes ~X. he term multiplying! is always psitive and hence is dissipative. Substituting N ~ frm Equatin (1) gives an expressin fr the mments acting n the bject as a functin f psitin and rtatinal speed. = W ~ RB, BB,1 1 +W ~ RB, BB,1 1 1 ~ ks ~f +k s Xcm ~ + ~ X cm ~X cm,! ~X ~ N,W ~ X cm ~ X cm (1) where k s and f ~ are the spring and set cnstants frm the translatinal dynamics, ~ k s is a 1 cnstant vectr relating trque t psitin, and is a scalar cnstant trque. Nte that nthing in the previus mathematics invlved the rientatin f the bject, and hence, while the bject rests n a particular set f supprts, trque n the bject is nt a functin f rientatin. his is very imprtant fr determining stable rientatins. Design f Velcity Fields A set f 9 cnstants quanties the mass-springdamper dynamics f the bject as it rests n a single set f supprts. Atypical prblem is t create a velcity eld (described by X and V) t prduce massspring-damper behavir with unifrm desired prperties ver the entire array. In particular, we specify an equilibrium psitin and return spring stinesses, and ensure rtatinal equilibrium at the translatinal equilibrium. he analysis relies n the fllwing assumptins: he crdinate rigin is at the desired equilibrium. he cells are arranged with mirrr-symmetry arund the crdinate axes. he bject als has mirrr symmetry. When the bject rtates cunterclckwise, mre cells under the bject lie in the rst and third quadrants than in the secnd and furth. Mre specically, P x i y i >..1 ranslatinal Cnstants Equatin (1), species the translatinal dynamics f the bject using cnstants: spring cnstants in k s and tw cnstant set frces in f ~. We cnsider nly the case where k s is a diagnal matrix, decupling x and y mtins f the bject. he diagnal terms in k s tend t pull the bject twards a central equilibrium. he -diagnal terms act as circulatry terms, mving the bject arund the equilibrium, and are nt helpful fr psitining the bject. Eliminating the -diagnal terms simplies the design prblem and imprves the rtatinal prperties f the eld. When the equilibrium psitin, Xcme ~, and return spring strengths, k sxx and k syy, are specied, the bject will mve t the equilibrium psitin with the dynamics f the mass-spring-damper system shwn in Figure. At equilibrium, f ~ =,k s Xcme ~, s, in eect, ~f is specied. he design prblem then becmes the prblem f determining wheel velcities V given their psitins (specied in X and equivalently in B) and the cnstants k s and f ~. Sectin dened the functinal relatinship frm V t k s and f ~. his sectin describes a methd t derive a suitable velcity matrix V frm k s and f ~. Equatin 1 denes the cnstants k s and f ~ as fx k sxx k sxy f y k syx k syy = W VB, BB,1 : (1)

5 ( x, y ) cme cme ( x,y ) Figure : With k sxy = k syx = the bject behaves like a mass-spring damper system. We rewrite this equatin in terms f the vectr (f length n) frmed by stacking the transpses f the tw rws f V ( V ~ x and V ~ y ). aking advantage f the symmetry f BB, the fllwing relatin hlds. f x k sxx k sxy f y k syx k syy =W cm cm ", # BB,1 B, BB,1 B B " ~V x ~V y # (18) his prduces a set f equatins and n unknwns. Slve fr the wheel speeds ( ~ V x and ~ V y ) requires the inversin f the previus set f equatins. his set f equatins is undercnstrained, s sme freedm in the slutin exists and further cnstraints are required. he Penrse pseud-inverse accmplishes this by minimizing the sum f the squares f the wheel speeds. After applying prperties f matrix transpses and inverses the pseud-inverse, B y,is B y B, BB,1 = B B : (19) herefre, we can slve fr the set f wheel speeds which will give an bject the desired dynamics expressed by the desired equilibrium and spring cnstants (with k sxy = k syx = t decuple the x and y mtins f the bject). " ~V x ~V y # = 1 B W B f x k sxx f y k syy () Since each rw f B is the vectr 1 x i y i, each cell's wheel speeds are cmputed independently. In the diagnal k s case, f x =,x cme k sxx, and f y =,y cme k syy, s the wheel speeds are V xi = k sxx (x i, x cme ) ; and (1) V yi = k syy (y i, y cme ) ; () which is a eld where the cells pint twards the equilibrium (fr negative k sxx and k syy ), with velcities f each cmpnent prprtinal t the perpendicular distance t the crrespnding axis (see Figure ). Nte that this is a discretized versin f the cntinuus elliptic eld described by Kavraki [].. Rtatinal Cnstants Sectin shwed that fr an an bject resting n a single set f supprts, the trque is nt a functin f rientatin. herefre, it is nt pssible t cnstruct a static velcity eld which will rient an bject mre precisely than its range f mtin which keeps it n a single set f supprts. he black rectangle in Figure demnstrates this free range f mtin. Lcally,wecan nly assure that the bject will be in rtatinal equilibrium when it is in translatinal equilibrium. Objects are riented as they change supprt frm ne cell t the next. here are then three cnsideratins fr the bject's rientatin: (i) rque is zer when translatinal frce is zer (at X ~ cm = X ~ cme ). (ii) When the bject rtates abut its equilibrium psitin, a change in supprts induces a restring trque. (iii) Given any starting psitin and equilibrium, the bject will eventually reach the desired psitin and rientatin. hese cnsideratins will be examined under the eld derived in Sectin.1. Withut lss f generality, the rigin f the crdinate system is placed at the desired equilibrium psitin. herefre, when the bject rtates abut its translatinal equilibrium, Xcm ~ = and f ~ =, and Equatin 1 reduces t =,! ~ X ~ N () which is a cnstant applied trque with linear damping. We must then have = fr the bject t be in cmplete translatinal and rtatinal equilibrium. he expressin fr frm Equatin 1 is = W 1, BB,1 B ~ R : () he vectr ~ R can be expressed in terms f the stacked velcityvectr. Furthermre, given ~ f = (due t the

6 chice f crdinate system) and k s is diagnal (by design), the cnstant-trque-term is = W 1, BB,1 B,y 1 x ,y n x n B B k sxx () k syy he rws f B are the nes vectr, the vectr f x psitins, and the vectr f y psitins. herefre, the terms prduced by multiplying B with ther vectrs and matrices frm sums f the x and y cmpnents f all the cell lcatins. Fr example, BB = P P P n P xi P yi xi xi xi y P P P i yi xi y i yi In terms f these sums, Equatin becmes = W 1 P P P n P xi P yi,1 P xi P xi Px i y i yi xi y i yi P P xi y i P xi y i xi y i : (), k syy,k sxx () In general, the trque resulting frm these cnstants evaluated at the equilibrium psitin is nt zer. Hwever, cnsider the case where the cells n which an bject rests are arranged symmetrically (mirrred in x and y) abut the crdinate axes. herefre, any term with dd pwers f x i r y i in a summatin will be identically zer. Fr example, in P x i y i, cells in the rst and furth quadrants cancel cells in the secnd and third quadrants, making P x i y i =. Similarly, P x i y i = and P x i y i =. herefre, the cnstant trque becmes identically zer such that the bject will be in rtatinal equilibrium when resting n a mirrr-symmetric set f supprts at the translatinal equilibrium. he symmetric arrangement f cells under the bject depends n the bject's shape and rientatin. Figure shws a rectangular bject in three rientatins at its equilibrium psitin with arrws at each cell indicating the magnitudes f the velcities at each cell. In this gure, we can see that bth the symmetrically riented bject (slid black line) and the slightly perturbed bject (dashed black line) have a symmetric set f frces, s d nt feel a trque. Hwever, the bject which has rtated enugh t change supprts (dtted black line) has a set f supprts which is symmetric abut the rigin (radially symmetric) rather Figure : Rtatin f tw bjects (black and grey) abut their equilibrium psitins. Black dashed lines represent the free (unrientable) range f mtin f the black bject. he grey bject des nt satisfy the \psitive rtatin" prperty. Nte that k sxx ;k syy < and (k syy, k sxx ) <. than abut the crdinate axes and will feel a trque. Equatin shws that fr a given arrangement f cells and a particular rientatin f bject, the directin f trque depends n the dierence k syy, k sxx. herefre, these cnstants determine the stability f rtatinal equilibrium. If k sxx = k syy there will be n trque when the bject rtates enugh t shift cells. If the bject is mirrr-symmetric itself, a strnger statement can be made abut the directin f rtatin. he resulting set f supprts will be radially symmetric at any bject rientatin such that fr every cell (x i ;y i ) there is a cell (,x i ;,y i ). hus, many fthe terms in Equatin becme zer. = W 1 n P P P x i Px i y i x i y i yi = 1 n,1 P xi y i, k syy,k sxx X x i y i, ksyy, k sxx : (8) A nal assumptin can be made that when the symmetric bject rtates cunterclckwise abut the equilibrium psitin, P x i y i >. his is ften true, since mre f the bject is in the rst and third quadrants, and mre rst and third quadrant cells (giving x i y i > ) are cvered. he black rectangle in Figure shws an bject with this prperty. Hwever, because f the discreteness f the array, sme bjects (fr example, the grey rectangle in Figure ) mayhave a negative P x i y i fr sme cunterclckwise rtatins. A particular bject can be checked fr \rientability"

7 n the array by analyzing it as it rtates abut equilibrium and t see which cells it cvers. Fr bjects with this \psitive rtatin", prperty, there will be a restring trque fr k syy, k sxx < with a stable rientatin. Nthing has been said s far fr trques n the bject when its psitin is nt at the equilibrium. assure prper rientatin, the bject must rst reach its equilibrium and then rient itself. Reaching the equilibrium is guaranteed regardless f rientatin and cell distributin since the bject's dynamics are that f a mass-spring-damper centered at the rigin. Once translatinal equilibrium is reached, supprt changes due t rtatin d nt aect the translatinal dynamics. Once psitin equilibrium is reached, a trque will be, applied which tends t rient the bject. Fr ksyy, k sxx <, a symmetric bject satisfying the psitive rtatin prperty will rtate until it is aligned with the crdinate axes. Cnclusins In this paper, a standard wheel velcity eld was methdically designed t prvide unifrm, arbitrary spring cnstants and equilibrium psitin ver the entire array regardless f which cells supprt the bject. he resulting eld, when nly the nn-circulatry spring cnstants are used, is an inward-pinting eld with each wheel's velcity prprtinal t its the crrespnding cmpnent f perpendicular distance t the equilibrium psitin. his eld was then analyzed fr its rtatinal equilibrium prperties. Interestingly, the trque n the bject is nt a functin f the rientatin because the supprting frces nly depend n the psitin f the center f mass. herefre, transitins frm cell t cell as the bject rtates abut its equilibrium rient the bject within the reslutin limits f the discrete array. Given the assumptin that the set f cells supprting the bject at equilibrium is mirrr-symmetric abut the crdinate axes, it was fund that a rtatinal equilibrium exists when the bject is at translatinal equilibrium. A ramicatin f this assumptin n a regular square-lattice array, such as the MDMS, is that the equilibrium psitin must be set either exactly n a cell r exactly midway between cells. Als, an bject can be riented nly t angles f n, fr n =;:::;. he trque develped by the inward-pinting eld was analyzed as the bject rtates abut equilibrium. It was determined that, as lng as the tw spring cnstants are nt equal, when the bject rtates and the supprts change, there will be a restring trque P n a symmetric bject with the prperty that x i y i > fr all the cells under the bject when the bject rtates cunterclckwise. his psitive rtatin prperty is an artifact f the discrete array and hlds fr many bject sizes and shapes. Furthermre, in the the limit f many cells spaced clsely tgether (e.g., a cntinuus array), this prperty always hlds fr symmetric bjects, as implied by Kavraki. he discrete case requires a simple analysis t check if an bject meets this assumptin. References [1] K.F. Bhringer, B.R. Dnald, R. Mihailvich, and N.C. MacDnald. A thery f manipulatin and cntrl fr micrfabricated actuatr arrays. In Prceedings. IEEE Internatinal Cnference n Rbtics and Autmatin, 199. [] K.Y. Gldberg. Orienting plygnal parts withut sensrs. Algrithmica: Special Issue n Cmputatinal Rbtics, 1:1{, August 199. [] L. Kavraki. Part rientatin with prgrammable vectr elds: w stable equilibria fr mst parts. In Prceedings. IEEE Internatinal Cnference n Rbtics and Autmatin, 199. [] J. Luntz, W. Messner, and H. Chset. Parcel manipulatin and dynamics with a distributed actuatr array: he virtual vehicle. In Prceedings. IEEE Internatinal Cnference n Rbtics and Autmatin, 199. [] J. Luntz, W. Messner, and H. Chset. Virtual vehicle: parcel manipulatin and dynamics with a distributed actuatr array. In Prceedings f SPIE vl. 1. Sensrs and Cntrls fr Advanced Manufacturing. Internatinal Sympsium n Intelligent Systems and Advanced Maufacturing., 199. [] J. Luntz, W. Messner, and H. Chset. Velcity Field Design n the Mdular Distributed Manipulatr System. In Prceedings f Wrkshp n the Algrithmic Fundatins f Rbtics., 1998.