Formation Control with Leadership Alternation for Obstacle Avoidance

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1 Pepnts of the 8th IFAC Wold Congess Mlano (Italy) August 8 - Septembe, Fomaton Contol wth Leadeshp Altenaton fo Obstacle Avodance Jose M. V. Vlca Maco H. Tea Vald Gass J. Depatment of Electcal Engneeng, Unvesty of São Paulo at São Calos, São Paulo, Bazl e-mal: {jmguelv, tea, vgass}@sc.usp.b Abstact: Ths pape deals wth leade-followng fomaton contol of a goup of wheeled moble obots (WMRs) when the leadeshp altenates among the obots of the goup. In the poblem pesented hee, fstly a scout obot s used to seach fo a path n a pevously unknown envonment wth statc obstacles. The path found by ths scout s used as a efeence fo the leade of the mult-obot fomaton. As the goup follows the tajectoy pesevng the fomaton, wheneve any obot n the fomaton s close than a lmt theshold dstance fom an obstacle, ths obot assumes the leadeshp. So the new leade must adequate ts poston to the efeence tajectoy, and, as a esult of the fomaton contol, the poston of the whole goup s also adjusted. Ths way, the whole goup navgates avodng collsons wth obstacles. The moble obots exchange the poston nfomaton among themselves accodng to a pespecfed communcaton dected gaph (dgaph). A dffeent dgaph s defned fo each obot as a leade. Smulaton esults ae pesented fo the fomaton contol appled on ths scenao. Keywods: Moble obots, unknown envonment, obstacle detecton, obstacle avodance, fomaton contol.. INTRODUCTION In ecent yeas, eseach n contol and coodnaton of multple moble obots has gown sgnfcantly. Tasks that may be too dffcult fo a sngle obot to pefom alone, can be pefomed moe effcently and cheaply by a goup of multple obots. Some examples of applcatons can be seen n exploaton, Kuppa et al. (), seach and escue, Whelan et al. (997), mappng of unknown locatons, Howad (5) and Mne (7), and tanspot o eposton of lage objects, Chamowcz et al. (4). A patcula poblem of mult-obot coodnaton s fomaton contol,.e., when a goup of moble obots has to keep a desed fomaton whle t moves along a collsonfee tajectoy. In geneal, thee ae thee mpotant appoaches to deal wth ths poblem: leade-followe, Desa et al.(), behavo-based, Tang et al.(6) and vtual leade appoach, Ghommam et al. (). In ths wok we addess the fomaton contol poblem usng the leadefollowe appoach. We can see n the lteatue some stateges that combne fomaton contol and tajectoy plannng fo evey moble obots n the goup, Yang et al. (7), Cuz and Caell (8) and Yang et al. (8). These woks also consde obstacle avodance when plannng the efeence tajectoy fo each obot, snce obstacle avodance s essental n applcatons such as navgaton, seach and escue n unknown envonments. Ths wok was suppoted by CNPq unde gant 47788/8-5 and 35753/9- Path plannng wth obstacle avodance n unknown envonment conssts n fndng a collson-fee path fom a gven ntal poston to a desed taget poston, Choset et al. (5) and LaValle (6). Senso systems such as sonas, cameas o lase measuement unts ae used to solve ths knd of poblem, see We and Zefan (5) and Cowan et al. (3). Howeve, the moton plannng fo multple obots wth obstacle avodance has a numbe of dffcultes and peculates. They ae elated to the necessty of takng nto account not only possble obstacles n the envonment, but also the movement of each obot n the goup. Theefoe, plannng s geatly smplfed f we plan the tajectoy fo one obot of the goup, usng eactve behavo to make the goup as a whole to avod obstacles. Ths pape poposes the use of a leade-followng fomaton contol wth leadeshp altenaton as a stategy fo a goup of obots to navgate whle keepng a specfc fomaton n an envonment wth statc obstacles. Hee we consde that the obstacles could be epesented by polygons, smooth cuves o some combnaton of cuved and lnea pats. Also, we consde that the envonment has suffcent space fo the navgaton of the moble obots n fomaton. We plan a collson-fee tajectoy fo the leade of the goup usng the TangentBug algothm. Consdeng a pe-specfed dstance of the obstacle, the obot that tansposes ths dstance assumes the leadeshp of the goup. The fomaton contolle appled n ths pape s descbed by Wllams et al. (5) and Matha et al. (7). Fo ths contolle, the moble obots exchange the poston nfomaton among themselves accodng to a pe-specfed Copyght by the Intenatonal Fedeaton of Automatc Contol (IFAC) 9

2 Pepnts of the 8th IFAC Wold Congess Mlano (Italy) August 8 - Septembe, communcaton dected gaph (dgaph) that s defned by the contolle desgne, see Fax and Muay (3). A dffeent dgaph s defned fo each obot as a leade. Ths pape s oganzed as follows: Secton descbes the algothm used to geneate the efeence tajectoy; Secton 3 descbes the fomaton contol, ts popetes and epesentatons; Secton 4 pesents the knematc model fo WMRs; and fnally, Secton 5 shows some smulaton esults.. REFERENCE TRAJECTORY GENERATION Ths secton pesents the algothm used to fnd a collsonfee path fom a gven ntal poston to a pedefned taget pont n an unknown envonment wth a fnte numbe of statc obstacles. It s consdeed a scenao n whch a scout obot s used to seach fo a path n a pevously unknown statc envonment. The path found s then used as a efeence fo the leade of a mult-obot fomaton. The scout moble obot that wll geneate the efeence tajectoy s equpped wth a lmted ange senso, whch s a lase measuement senso (LMS) wth 8 degees scannng ange. The stategy appled s based on the TangentBug algothm. It explots ange data fom LMS and guaantees that the moble obot eaches the taget f possble o poves that the taget cannot be eached.. TangentBug algothm The Bugs algothm solves plannng poblems usng a stategy based on maze exploaton technques. The TangentBug algothm was poposed n Kamon et al. (996). It pesents bette pefomance than classcal Bugs algothm, and t s used fo tajectoy plannng due to ts staghtfowad mplementaton, see Kamon et al. () and We and Zefan (5) TangentBug s based on two behavos, moton towads taget and followng obstacle bounday. The swtchng between behavos depends on the dstance fom the poston of the moble obot x to the taget T, whch s epesented by d(x,t). In the moton towads taget behavo, d(x, T) deceases monotoncally whle the followng obstacle bounday behavo attempts to escape fom local mnmum of d(x,t). The moble obot constucts a local tangent gaph (LTG) usng the eadngs fom the ange senso. It s constantly updated to be used by the moble obot to decde the next moton. Below, a summay of the algothm s pesented. () Move towad T along the locally optmal decton, whch s the decton along the shotest path to the taget accodng to the cuent LTG, untl one of the followng events occus: The taget T s eached. Stop. A local mnmum of d(x,t) s detected. Go to step. () Choose a decton to follow the obstacle bounday decton. Move along the obstacle bounday usng the LTG whle ecodng d followed (T), the mnmal dstance along the bounday of the followed obstacle to the taget, and d eached (T), the mnmal dstance wthn the vsble envonment to the taget, untl one of the followng events occus: The taget T s eached. Stop. The leavng condton d eached < d followed holds. Go to Step. The moble obot completes a loop aound the obstacle. The taget T s uneachable. Stop. 3. FORMATION CONTROL WITH LEADERSHIP ALTERNATION 3. Fomaton Contol The fomaton contol pesented n ths secton uses the efeence tajectoy povded by the algothm n Secton. The moble obots acheve and mantan pe-specfed elatve postons and oentatons wth espect to each moble obot n fomaton followng ths efeence, see Matha et al. (7). The postons and veloctes of N moble obots n fomaton wth espect to the netal coodnates can be epesented by: ẋ = A veh x +B veh u, =,...,N x R 4, whee the vecto x = [x,ẋ,y,ẏ ] T epesents the poston of th moble obot and the devatves, and u savectoofcontolnputs.inthsappoachthewmrs move n two dmensons and the matces A veh and B veh ae gven by a A veh = 4, B veh =, a 4 whee a j s elated to the desed tajectoy fo the fomaton, and the vaaton of these paametes can poduce a vaable tajectoy fo the leade moble obot. The zeos n the columns one and thee of A veh guaantee the convegence of the WMRs nto fomaton (see Laffeee et al. (4) Poposton 3. and Veeman et al. (5) Poposton 4.). The vecto x = (x,x,...,x N ) T descbes the combnaton of all states of N moble obots. The vectos of poston and velocty ae defned as: x p = [ ),..., ) N ] T, x v = [(x v ),...,(x v ) N ] T, whee ) = [x,y ] T and (x v ) = [ẋ,ẏ ] T. Hence, x can be wtten as: ( ( x = x p +x ) v, ) whee denotes the Konecke poduct. In ths wok, a fomaton s defned by the vecto ( h = h p R ) 4N, whee h p = [ (h p ) T,...,(h p ) T N] T s the vecto of the fomaton postons wth espect to the netal coodnates, and (h p ) = [h x,h y ]. 9

3 Pepnts of the 8th IFAC Wold Congess Mlano (Italy) August 8 - Septembe, The N moble obots ae n fomaton h at tme t f thee exst vectos q,w R n such that ) (t) (h p ) = q and (x v ) (t) = w, fo =,...,N. The WMRs convege to fomaton h f thee exst R n -valued functons q(.),w(.), such that ) (t) (h p ) q(t) and (x v ) (t) w(t), as t, fo =,...,N. Fgue llustates ths concept of fomaton. Y h x h x 3 x ( t) x ( t) qx( t) qx( t) Poston at tme t Fg.. Thee WMRs n Aow Tp Fomaton wth leadeshp altenaton. The topology of communcaton among moble obots s featued by a dgaph Γ whch captues the communcaton lnks among them (see Laffeee et al. (5) fo defnton of dgaph). Each vetex epesents a moble obot and a dected edge epesents communcaton fom one obot to the othe. The WMR that eceves nfomaton fom othe WMR, adjusts ts own state based on ths nfomaton. These two obots ae consdeed neghbos. The leade of the fomaton does not eceve nfomaton fom any obot, t only sends ts pose to ts neghbos. Fo each WMR, J denotes the set of ts neghbos and the contol nputs u ae functons of x j x and h j h fo each j J. A natual way to combne the elatve nfomaton when thee s a leade n fomaton, aound whch the othe WMRs must adjust the movements (as done n Fax and Muay. (3)), s to defne output functons z computed fom an aveage of the elatve dsplacements (and veloctes) among WMRs as follows: ((x h ) (x j h j )), f J, z = J j J, othewse, fo =,...,N, whee J ndcates the numbe of neghbos fo the obot. The coespondng output vecto z can be wtten as z = L(x h) whee L = L Γ I 4 and L Γ s the dected Laplacan matx of the communcaton gaph Γ, see Wllams et al. (5). 3 X () Collectng the equatons fo all WMRs nto a sngle statespace system, we obtan: ẋ = Ax+Bu, z = L(x h), whee A = I N A veh and B = I N B veh. A decentalzed contol law appled to all moble obots, s defned by a feedback matx F of the fom F = I N F veh. The equatons fo all WMRs ae gven by: ẋ = I N A veh x+l Γ B veh F veh (x h). () Note that all WMRs wll convege to the gven fomaton f the dected Laplacan matx L Γ has at least two zeo egenvalues, see Laffeee et al. (5). In ode to obtan thecontolle,thefeedbackmatxf veh canbeepesented as follows: [ ] f f F veh =. f f Necessay and suffcent condtons fo the system () to convege to fomaton ae obtaned wth f <, f <, see Wllams et al. (5). The equatons fo all WMRs when they ae followng a specfed path ncludng both the coopeatve and tackng contolle, ae gven by ẋ = Ax+BF(x h)+k tack e tack, (3) e tack = ef x, (4) whee e tack s the path followng eo, ef s the efeence tajectoy, and K tack a vecto wth the tackng gans. Note that the stablty of the fomaton consdeng leadeshp altenaton s guaanteed fo the appoach pesented n ths pape by Wllams et al. (5). 3. Leadeshp Altenaton Stategy The leadeshp altenaton stategy s based on eactve contol and s used to obstacle avodance, see fo nstance Chamowcz et al. (4). We consde that the WMRs have ange sensos that pemt knowng the dstance to the obstacle. Hence, when one of the WMRs s close than a dstance theshold fom an obstacle, ths WMR assumes the leadeshp of the fomaton. Then t adjusts ts poston to tack the planned tajectoy. As a esult of the fomaton contol, the whole goup adjusts ts poston as well. Thus the leadeshp eactvely altenates so that the goup may avod collsons wth the obstacles. 4. WHEELED MOBILE ROBOT MODEL In ths secton the knematc model of the WMRs of the fomaton s pesented. We consde that the each WMR s a dffeental wheeled moble obot. The geomety of the obot s shown n Fg., whee (X,Y) s the netal coodnate system; (X,Y ) s the local coodnate system; P c s the cente of mass; a s the obot length; b s the dstance between the actuated wheel and the axs of symmety; s the adus of the actuated wheel; θ R and θ L ae the angula dsplacement of the ght and left wheels, espectvely. 9

4 Pepnts of the 8th IFAC Wold Congess Mlano (Italy) August 8 - Septembe, X a L θ L ψc X q d = [ ] θd θ l d = [ ][ ] / b/ v d / b/ ω d, () whee θ d and θ d l ae desed veloctes of the ght and left wheels, espectvely. xc Z Fg.. Wheeled Moble Robot. b P c b yc Moto, encode and actuated θ R wheel The WMRs of the fomaton pesent thee knematc constants, the fst mposes that the obot does not have lateal movement and the othe two mpose that the actuated wheels do not slp, see Coelho and Nunes (3). Defnng a coodnated wdespead q = [θ R,θ L ] T and q = [x c,y c,ψ c,θ R,θ L ] T, whee =,...,N, and N s the numbe of WMRs of the fomaton, the knematc equaton of the WMR s gven by: whee q (t) = S(q ) q (t), (5) R Y Y /cosψ c /cosψ c /snψ c /snψ c S(q ) = /b /b. The knematc-based contolle poposed by Kanayama et al. (99) povdes the desed wheel veloctes, such that WMRs tack the efeence tajectoes geneated by the fomaton contol. Consde that the eo q e = [x e,y e,ψ e ] T between the efeence poston P = [x,y,ψ ] gven by (3) and the actual poston of the WMRs n the fomaton P c = [x c,y c,ψ c ] has the followng fom: x e = cosψ c (x x c )+snψ c (y y c ), (6) y e = snψ c (x x c )+cosψ c (y y c ), (7) ψ e = ψ ψ c, (8) whee (x,y ) = q s the efeence tajectoy and ψ = actan(ẏ /ẋ ). Hence, the lnea (v d ) and angula ) desed veloctes of the WMRs ae gven by: (ω d v d = v cos(ψ e )+K x x e, (9) ω d = ω +v (K y y e +K ψ snψ e ), () whee K x,k y,k ψ ae constants to be defned by the desgne.theefeenceveloctesv andω aecomputed as: v = (ẋ ) +(ẏ ) and ω = ψ. () The elatonshp between the desed angula veloctes of the wheels, q d, and the desed lnea and angula veloctes of the obot s gven by: 5. SIMULATION RESULTS The smulaton of sx WMRs n fomaton has been pefomednmatlab R.FoeachWMR,weconsdethepaametes of the e-puck moble obot: m c =.5 kg, a =.7 m, b =.65 m, = 4. m. Fgue 3 shows the block dagam of the developed fomaton contol stategy. q Fomaton Contol T q. P c Leade k q. q T Knematc-based Contol Computaton of Poston Eo q e MOBILE ROBOT Knematc Model Dead-Reckonng Knematcal Tackng Contol Law Leadeshp Altenaton Leade Leade Leade N Collson-Fee Tajectoy fo the Leade d T d Fg. 3. Detaled block dagam of contol stateges wth leadeshp altenaton. Fgue 4 shows the sequence of fou dgaph used to each one of the obots when the leadeshp s altenated along the tajectoy. Note that n Fg. 4, the black ccle coesponds to the leade and the WMRs must keep the followng fomaton: (h p ) = [, ]l, (h p ) = [, ]l, (h p ) 3 = [, ]l, (h p ) 4 = [,.5]l, (h p ) 5 = [, ]l, (h p ) 6 = [,.5]l. whee l =.3 m. h h h h Fg. 4. Dgaph fo each leade moble obot of the fomaton. It s consdeed that each WMR has a thee ange sensos wth maxmum ange of.5 m whch ae used to detect the dstance to an obstacle. It s also assumed that the obot has the ablty to dstngush f the data fom the ange sensos efe to an obstacle o anothe obot. In ths smulaton, a obot assumes the fomaton leadeshp when h h h h 93

5 Pepnts of the 8th IFAC Wold Congess Mlano (Italy) August 8 - Septembe, t s less than.3 m fom an obstacle obseved though ts ange sensos. The WMRs wll be named accodng to the poston n the fomaton,.e, Leade s the moble obot n the postonh andsofoth.thefeedbackmatxf veh appled to all WMRs s gven by: F veh = [ ] 3. Fo the knematc-based contolle, the gans ae defned as K x =.45, K y =, and K ψ =.. The values of these gans wee chosen empcally to obtan a fast esponse and velocty values wthn lmt veloctes of the moble obot, whch ae υ max = m/s and ω max = ad/s. The efeence tajectoy fo the leade moble obot was geneated by the algothm TangentBug. It was smulated n MATLAB R consdeng that the maxmum ange of lase measuement senso (LMS) s 3 m. Fgue 5 shows the efeence tajectoy n an unknown envonment. y (m) ) ) ) 3 ) 4 ) 5 ) 6 Intal Poston Taget x (m) Fg. 6. Tajectoy fo sx moble obots avodng obstacle..5 y (m) Walls Refeence Tajectoy Intal Poston Taget x (m) Fg. 7. Leadeshp altenaton along the tajectoy. Fg. 5. Collson-fee tajectoy. The efeence tajectoes, x and y, ae gven by (3) wth the ntal condtons (x c,y c,α c ) = (,7/3,)l, (x c,y c,α c ) = (,5/6,)l, (x c3,y c3,α c3 ) = (, /3,)l, (x c4,y c4,α c4 ) = (,7/3,)l, (x c5,y c5,α c5 ) = (,5/6,)l, (x c6,y c6,α c6 ) = (, /3,)l. Fgue 6 shows the tajectoy fo all WMRs and the fomaton shape (dotted lne). Note that the obots do not collde wth any obstacle along the tajectoy and that they each the desed fomaton and mantans the shape of the fomaton. Fgue 7 shows the eactve contol fo the fomaton, whee the leadeshp s assumed by the moble obot that s closest to an obstacle along the tajectoy. The leade s epesented by a black ccle, the ange sensos ae epesented by the gay lne, and the gay ccle epesents the moble obot who has aleady been the leade and the whte ccle s the obot that stll has not been the leade. Note that the ange sensos do not show when a moble obot s detected. The lnea and angula veloctes fo all WMRs ae shown n Fgues 8 and 9, espectvely. Note that n both fgues, the velocty values ae less than the lmt velocty, also n Fg. 9, the angula velocty values pesent an abupt oveshoot that occus when leadeshp has been assumed by othe obot and t must avod the obstacle deceasng the lnea velocty and nceasng the angula velocty. v (m/s) v (m/s) t (s) Fg. 8. Lnea veloctes fo sx moble obots. v v v 3 v 4 v 5 v 6 94

6 Pepnts of the 8th IFAC Wold Congess Mlano (Italy) August 8 - Septembe, w (ad/s) w (ad/s) Fg. 9. Angula veloctes fo sx moble obots. t (s) 6. CONCLUSIONS Ths pape poposed a method fo fomaton contol of a goup of moble obots wth leadeshp altenaton based on eactve contol fo avodng obstacles of an unknown envonment. By combnng the fomaton contol wth leadeshp altenaton and the algothm TangentBug t was shown how to avod obstacles whle the obots keep the fomaton. The effectveness of ths appoach s demonstated though smulaton esults. The man esults of ths wok ae the leadeshp altenaton to avod obstacles usng eactve contol and applcaton of the efeence tajectoy plannng n unknown envonment fo only one obot that educe the computatonal complexty fo ths knd of poblem. In futue woks, the poblem of changng the fomaton shape and fault toleant contol of the WMRs n fomaton wll be consdeed. REFERENCES Chamowcz, L., Kuma, R.V., and Campos, M.F.M. (4). A mechansm fo dynamc coodnaton of multple obots. Autonomous Robots, 7, 7. Choset, H., Lynch, K.M., Hutchnson, S., Kanto, G., Bugad, W., Kavak, L.E., and Thun, S. (5). Pncples of Robot Moton: Theoy, Algothms, and Implementaton. MIT Pess. Coelho, P. and Nunes, U. (3). Le algeba aplcaton to moble obot contol: A tutoal. Robotca,, Cowan, N., Shakena, O., Vdal, R., and Saty, S. (3). Vson-based follow-the-leade. In Poc. IEEE/RSJ Int. Conf. Intellgent Robots and Systems, volume, Cuz, C.D.L. and Caell, R.(8). Dynamc model based fomaton contol and obstacle avodance of mult-obot systems. Robotca, 6(3), Desa, J., Ostowsk, J., and Kuma, V. (). Modelng and contol of fomatons of nonholonomc moble obots. IEEE Tansacton on Robotcs and Automaton, 7(6), Fax, J. and Muay, R. (3). Infomaton flow and coopeatve contol of vehcle fomatons. IEEE Tansactons on Automatc Contol, 49(9), w w w 3 w 4 w 5 w 6 Fax, J. and Muay., R.(3). Infomaton flow and coopeatve contol of vehcle fomatons. IEEE Tansactons on Automatc Contol, 49(9), Ghommam, J., Mehjed, H., Saad, M., and Mnf, F. (). Fomaton path followng contol of uncycletype moble obots. Robotcs and Autonomous Systems, 58(5), Howad, A. (5). Mult-obot smultaneous localzaton and mappng usng patcle fltes. In In Robotcs: Scence and Systems, 8. Kamon, I., Rmon, E., and Rvln, E.(). Range sensobased navgaton n thee-dmensonal polyhedal envonments. Intenatonal Jounal of Robotcs Reseach, (), 6 5. Kamon, I., Rvln, E., and Rmon, E. (996). A new ange-senso based globally convegent navgaton algothm fo moble obots. In Intenatonal Confeence on Robotcs and Automaton. IEEE, Mnneapols, Mnnesota. Kanayama, Y., Kmua, Y., Myazak, F., and Noguch., T. (99). A stable tackng contol method fo an autonomous moble obots. n Poc. IEEE Intenatonal Confeence on Robotcs and Automaton, Kuppa, H., Fox, D., Bugad, W., and Thun, S. (). A pobablstc appoach to collaboatve multobot localzaton. Auton. Robots, Laffeee, G., Caughman, J., and Wllams, A. (4). Gaph theoetc methods n the stablty of vehcle fomatons. Poceedng of Amecan Contol Confeence, Laffeee, G., Wllams, A., Caughman, J., and Veeman., J. (5). Descentalzed contol of vehcle fomatons. Systems and Contol Lettes, 54(9), LaValle, S.M. (6). Plannng Algothms. Cambdge Unvesty Pess. Matha, K., Laffeee, G., and Ttenso, T. (7). Coopeatve contol of uav platoons - a pototype. In Euo UAV 7 Confeence and Exhbton. Pas, Fance. Mne, D. (7). Swam obotcs algothms: A suvey. Tang, H., Song, A., and Zhang, X.(6). Hybd behavo coodnaton mechansm fo navgaton of econnassance obot. In Intenatonal Confeence on Intellgent Robots and Systems. Bejng - Chna. Veeman, J., Laffeee, G., Caughman, J.S., and Wllams, A. (5). Flocks and fomatons. Jounal of Statstcal Physcs, (5-6), We, S. and Zefan, M. (5). Smooth path plannng and contol fo moble obots. In IEEE Int. Conf. On Netwokng, Sensng and Contol. Tucson, Azona. Whelan, G., Jennngs, J., and Evans., W. (997). Coopeatve seach and escue wth a team of moble obots. IEEE Int. Conf. Advanced Robotcs, 93. Wllams, A., Laffeee, G., and Veeman, J. (5). Stable motons of vehcles fomatons. Poceedngs of the 44a. IEEE Confeence on Decson e Contol, and the Euopean Contol Confeence, 5. Yang, T.T., Lu, Z.Y., Chen, H., and Pe, R. (8). Fomaton contol and obstacle avodance fo multple moble obots. Acta Automatca Snca, 34(5), Yang, T., Lu, Z., Chen, H., and Pe, R. (7). Robust tackng contol of moble obot fomaton wth obstacle avodance. Jounal of Contol Scence and Engneeng, 7(),. 95

Scalars and Vectors Scalar

Scalars and Vectors Scalar Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg