PROFILING FINGERS, A MEANS OF IMPROVING THE PRECISION OF THE GRIPPERS

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1 SISOM 004, BUCHAREST, 0-1 May PROFILING FINGERS, A MEANS OF IMPROVING THE PRECISION OF THE GRIPPERS Hia PANAITOPOL, Cnstantin UDREA Plitehnia Univesity f Buhaest, Meanial Faulty Engineeing, Mehatnis Dept., Abstat: F the gippes with tw tative finges and pismati jaws, the gasping f the ylindial bjets with big vaiatins f diametes leads t unaeptable es f gasping. Nevetheless, thee is a pssibility t utilize finges with shaped ative sufae in de t ealize an e f gasping theetially null. At the same time, a bette kinetstati behavi, by mpaisn with the utilizatin f the pismati jaws, is ensued. Nevetheless, the utilizatin f the pfiled finges is limited: n the ne hand, beause f the tehnlgial easns; n the the hand, beause f the equiements nening the featues f the elements whih belng t the stutue f the gippe. F the tw finge gippes, the usual fm f the jaws is the pismati ne. In this ase, the tansvesal gasping f the ylindi manipulated bjet (MO), whih has geat vaiatins f diamete, leads t unaeptable es f enting. One f the pssibilities f eliminating the es is epesented by the utilizatin f the pfiled jaws. The bjets an be manipulated in a wide ange f diametes, fm D min (fig. 1, a) t D (fig. 1, b). The jaws have the ative sufaes 3 and 5 pfiled in suh a way as t ealize an e f gasping theetially null. Fm a theeti pint f view, the speifi pblem f these gippes nsists in the synthesis f the pfiles f the jaws whih will we appah futhe n. The wking sufaes f the jaws will be desibed n a plane sufae as envelpe iles f diffeent diametes espnding fm the dimensinal pint f view t the MO. In de t failitate the mathematial expessin, we will nside the finge t be fixed/immutable and the MO f diffeent mbile adii R (fig. ). Keeping the gasping ente O will mean maintaining nstant the dimensin l. Theefe, the pla dinates f the entes f the iles ae: x= l s (1) y = l sin () The expessin f the uent adius R depending n the pla angle will be:

2 Hia PANAITOPOL, Cnstantin UDREA 356 R = R ( R R ) (3) min In the elatin (3), R and R min ae the exteme adii f the MO and is the utmst angle f sillatin Fig. 1. Finges with pfiled jaws. haateizing the finge in between the psitins espnding t the exteme states. This angle will be taken int nsideatin. Its value influenes the kinetstati aspet, as it will be demnstated futhe n. If the fllwing ntatin is used: k = R R / (4) the expessin f the uent adius bemes: ( ) min R = R k The equatin f the family f iles whih will be envelped is witten: ( x x) + ( y y) = R (6) (5) The abve elatin is deivated with and the fllwing elatin is btained: x x x y y y = R R ( ) ( ) The elatins (6) and (7) fm a system whih pemits detemining x and y : R x x R x y x y = ± + R x + y R y y R y x x y = + R x + y (7) (8) (9) The elatins (8) and (9) epesent the paameti expessin f the envelpe uves. In the patiula ase f the pfiles f the jaws, in espndene with the elatins (1), () and (5), it will be btained: x = l sin (10) y = l s (11) R = k (1) The diffeentials (10), (11) and (1) intdued in (8) and (9) lead t: R x1, = x ( k sin ± E s) (13) l R y1, = y ( k s± E sin ) (14) l In the elatins (13) and (14) ae nisely endeed: with the sign + the dinates f the pint R 1 f the envelpe uve C 1 (fig. 3) and with the sign - the dinates f the pint R f the the envelpe uve C, with E being nted: Fig.. The btaining f the pfile C f the jaw as envelpe uves. E = l k The kinetstati alulatin inludes, at the ning, the deteminatin f the inteatin fes between the MO and the jaws, knwing the mment M ealized by the atuat mehanism f the gippe at the level f the finge atiulatin. The lamping fe an be dedued fm the value f the mment M. The fe is pependiula n the dietin OO : (15)

3 357 Pfiling finges, a means f impving the peisin f the gippes Q = M / l (16) Beause f the symmety f the geneatin f the ntat sufaes C 1 and C, the tiangle R 1 R O is isseles. Theefe, the fes N 1 and N ae symmetial elatively t the dietin f the fe Q. Can be witten: Q N = N1 = N = (17) s β Taking int nsideatin the elatin (16), it esults: M N = (18) l s β In de t alulate the angle β, the axial system an be hanged t OX 0 Y 0 Z 0, whih means binging the finge in the psitin in whih it gasps the MO fm the uent adius R. The tansfmatin matix between the tw axial system is: s sin 0 T = sin s 0 (19) The vets f the psitin f the pint R 1 and R expessed in the new axial system ae alulated with the elatins: x1 x1 y 1 = T y1 (0) 0 0 x x y = T y (1) 0 0 Fig. 3. Gemetial elements f the alulatin f the angle β. By the substitutins in adane with (1), (), (13) and (14) it esults: x y E R = l l () k R = y = l (3) E R = l+ l (4) 1 x 1 It an be witten: sβ= y R And afte the emaking f the alulatin with the elatins (4) and (3) it bemes:

4 Hia PANAITOPOL, Cnstantin UDREA 358 k R sβ= = l R l min (5) The expessin (18) f the ntat fe bemes: N = (6) M ( R R ) An adimensinal fat an be intdued with the bjet f sme mpaative analysises: H N l min l n = = (7) M ( R Rmin ) F the patial dimensins f the atuating mehanism, it stats fm the MO with the imum weight G whih nmally espnds t D. The fes N ne an be alulated with kinetstati easns, in de t avid the tansvesal gliding: N ne kal G = 4 µ (8) Then, the mment M ne is alulated with the elatin (6). F the net alulatin f the pfile f the jaws and f thei gaphi illustatin a pgamme mminfas was ealized. The listing f this pgamme is endeed futhe n. In the fist pat {A} the alulatin f the dinates f the pints R 1 and R is ealized n the basis f the elatins (1), (), (13) and (14). At the level {B} and {C} the pfiles f the sufaes C 1 and C espetively ae dawn. At the level {B*} ae dawn the iles whih epesent the psitins f the envelped MO. pgamme mminfas; uses t,mate,gaph; va,d,dmin,,min,l,t,t,int,x,y,x1,y1,x,y,k,intx,inty,kdes :eal; i,gd,gm,n :intege; ls; n:=10; kdes:=1.5; wite(' D='); eadln(d); wite(' Dmin='); eadln(dmin); wite(' Teta_='); eadln(t); wite(' l='); eadln(l); t:=t*pi/180; l:=10; :=d/; min:=dmin/; k:=(-min)/t; int:=sqt(l*l-k*k); {A} f i:=1 t n+1 d t:=t*(i-1)/n; x:=l*s(t); y:=l*sin(t); :=-k*t; intx:=x1; inty:=y1; x1:=x-/l*(k*sin(t)+s(t)*int); y1:=y-/l*(-k*s(t)+sin(t)*int); x:=x-/l*(k*sin(t)-s(t)*int); y:=y-/l*(-k*s(t)-sin(t)*int); witeln(' i=',i,' D=',*:6:3,' x1=',x1:6:3,' y1=',y1:6:3,' x=',x:6:3,' y=',y:6:3); end;

5 359 Pfiling finges, a means f impving the peisin f the gippes hn:=t*l/(d-dmin); witeln( Hn=N*l/M=,hn:6:3); eadln; initgaph(gd,gm,':\bp\bgi'); setbkl(white); setl(8); line(100,40,und(100+kdes*50),40); line(100,40,100,und(40-kdes*100)); f i:=0 t d line(100,und(40-kdes*50*i),105,und(40-kdes*50*i)); f i:=0 t 5 d line(und(100+kdes*50*i),40,und(100+kdes*50*i),35); uttextxy(90,50,'0'); uttextxy(und(90+kdes*50),50,'50'): ` uttextxy(und(105+kdes*50),35,'x'); uttextxy(70,und(45-kdes*100),'100'); uttextxy(95,und(30-kdes*100),'y'); uttextxy(10,380,'gripper WITH PROFILED JAWS - THE TRACING OF THE PROFILE'); uttextxy(10,400,'d=160mm; Dmin=10mm; l=10mm; Teta=50 '); t:=0; :=-k*t; x:=l*s(t);y:=l*sin(t); x1:=x-/l*(k*sin(t)+s(t)*int); y1:=y-/l*(-k*s(t)+sin(t)*int); mvet(100+und(kdes*x1),40-und(kdes*y1)); {B} f i:=1 t n+1 d t:=t*(i-1)/n; x:=l*s(t); y:=l*sin(t); :=-k*t; intx:=x1;inty:=y1; x1:=x-/l*(k*sin(t)+s(t)*int); y1:=y-/l*(-k*s(t)+sin(t)*int); setlinestyle(0,0,3); line(100+und(kdes*intx),40-und(kdes*inty),100+und(kdes*x1),40-und(kdes*y1)); {B * } setlinestyle(0,0,1); ile(100+und(kdes*x),40-und(kdes*y),und(kdes*)); delay(00); end; t:=0; :=-k*t; x:=l*s(t); y:=l*sin(t); x:=x-/l*(k*sin(t)-s(t)*int); y:=y-/l*(-k*s(t)-sin(t)*int); mvet(100+und(kdes*x),40-und(kdes*y)); {C} f i:=1 t n+1 d t:=t*(i-1)/n; x:=l*s(t); y:=l*sin(t); :=-k*t; intx:=x; inty:=y; x:=x-/l*(k*sin(t)-s(t)*int); y:=y-/l*(-k*s(t)-sin(t)*int); setlinestyle(0,0,3); line(100+und(kdes*intx),40-und(kdes*inty),100+und(kdes*x),40-und(kdes*y)); delay(00); end; eadln; lsegaph; end. In de t ffe an exemplifiatin, we un the pgamme with the nete dates R = 80 mm; R min = 5 mm; =50 0 ; l = 10 mm. In the table 1 the fllwing date ae intdued: the uent diamete f the envelped ile and the dinates f the pints R 1 and R, f 11 pints f alulatin fm the wking field.

6 Hia PANAITOPOL, Cnstantin UDREA 360 Table 1 i D x 1 y 1 x y ,168 57,96 175,83 57, ,613 57, ,43 66, ,419 58, ,767 74, ,491 60, ,014 81, ,75 6, ,306 86, ,011 65,766 1,775 90, ,36 69,495 11,544 93, ,80 73,955 10,73 95, ,06 79,135 93,410 97, ,534 85,015 84,691 97, ,148 91,554 76,634 96,900 The value f the ntat fe fat (7) depends n the adpted angle. In this ase, it btains: H n = 0,698. This value is batte than the ne btained in the situatin when the MO was manipulated with lassi finge with pismati jaws. F the angle f the pism f 10 0 : H n = 1/ (*s30 0 )=0,433. The fm f the ntat sufaes f the jaws, as well as the geneatin mdel ae endeed in fig. 4, ealized by means f the abve pgamme. Fig. 4 Taing the pfiles f the jaws by means f the pgamme mminf REFERENCES 1. *** Dextus Rbt Hands, Spinge-Velag, Belin, 1990;. H., PANAITOPOL, GRIPPERS, Dissetatin f Dt f Siene, Plitehnia Univesity f Buhaest, 1999.