工业机器人手臂的静态平衡英文文献

1、mECHnmsm nno mnonne theorvPERGAMONMechanism and Machine Theory 35 (2000) 1287 1298 locate mechmtThe static balancing of the industrial robot armsPart I: Discrete balancingIon Simionescu1. IntroductionThe mechanisms of manipulators and industrial robots constitute a special category of mechanical sys

2、tems, characterised by big mass elements that move in a vertical plane, with relatively slow speeds. For this reason the weight forces have a high share in the category of resistance that the driving system must overcome. The problem of balancing the weight forces is extremely important for the play

3、-back programmable robots, where the human operator must drive easily the mechanical system during the training periodGenerally, the balancing of the weight forces of the industrial robot arms results in the decrease of the driving power. The frictional forces that occur in the bearings are not take

4、n Corresponding author.E-mail cuklress: simion(<? form.resist.pub.ro (I. Simionescu).0094-114X.00/S see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0094-1 14X(99)00067-1, Liviu CiupituMechanical Engineering Department. POLI TE1! NIC A University oj Bucharest. Splaiu!

5、 hidependeniei 3/3, RO-77206,Bucharest 6. RomaniaReceived 2 October 1998; accepted 19 May 1999AbstractThe paper presents some new constructional solutions for the balancing of the weight forces of the industrial robot arms, using the elastic forces of the helical springs. For the balancing of the we

6、ight forces of the vertical and horizontal arms, many alternatives arc shown. Finally, the results of solving a numerical example are presented 0 2000 Elsevier Science Lid. All rights reservedKeywords: Industrial robot; Static balancing; Discrete balancing1. Simionescu, L. Ciupitu ； Mechanism and Ma

7、chine Theory 35 (20(H) 1287-1298#into consideration because the frictional moment senses depend on the relative movement senses.In this work, some possibilities of balancing of the weight forces by the elastic forces of the cylindrical helical springs with straight characteristics are analysedThis b

8、alancing can be made discretely, for a finite number of work field positions, or in continuous mode for all positions throughout the work field. Consequently, the discrete systems realised only an approximatively balancing of the arm.The use of counterweights is not considered since they involve the

9、 increase of moving masses, overall size, inertia and the stresses of the components.2. The balancing of the weight force of a rotating link around a horizontal fixed axisThere are several possibilities of balancing the weight forces of the manipulator and robot arms by means of the helical spring e

10、lastic forces.The simple solutions are not always applicable Sometimes an approximate solution is preferred leading to a convenient alternative from constructional point of viewThe simplest balancing possibility of the weight force of a link 1 (the horizontal robot arm, for example) which rotates ar

11、ound a horizontal fixed axis is schematically shown in Fig. 1. A helical spring 2. joined between a point A of the link and a fixed B one, is used. The equation that expresses the equilibrium of the forces moments 1. which act lo the link L is(ni OG cos (pj + 也)g + 代。=0, / = 1 ，6.(1)where the elasti

12、c force of the helical spring is:= Fak(AB-andFig. 11. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-12981289XbYa-Xa Yba= 7bcos(p( sin ® sin (pj cos <PjAB = y)(XA - Ya -加2.4 =沽©The gravity centre G? of spring 2 is collinear with pairs centres A and B.The stiffness

13、coefiicient of the spring is denoted by k、ni is the mass of the link 1, is the mass of the helical spring 2, and g represents the gravity acceleration magnitude.Thus, the unknown factors: .口小F()and k may be calculated in such a way thatthe equilibrium of the forces is obtained for six distinct value

14、s of the angle 申卜 The movable coordinate axis system xOy attached to the arm 1 was chosen so that the gravity centre G is upon the Ox axis. The coordinalcs xa and yA defined the position of point A of the arm 1.In the particular case, characterised by yA =址=/()=厲=(),the problem allows an infinite nu

15、mber of solutions, which verify the equation:L _ (耳 OS + 山M4)g= 'for any value of angle cp.Since in this case. F、= k A B (see line 1, Fig. 2). some difliculties arise in the construction of this system where it is nol possible to use a helical extension spring. The compression spring, which has

16、to correspond to the calculated feature, must be prevented against buckling Consequently, the friction forces that appear in the guides make the training operation more difficult.Even in lhe general case, when vu/0 and X工0. results a reduced value of the initial length /o of lhe spring, correspondin

17、g to lhe forces F()= (). The modification of the straight characteristic position to the necessary spring for balancing (line 2, Fig. 2), i.e. to obtain an acceptable initial length /()from lhe constructional point of view, may be achieved by replacing the fixed point B of spring articulation by a m

18、ovable one In other words, the spring will be articulated with its B end of a movable link 2, whose position depends on that of lhe arm 1.Link 2 may have a rotational motion around a fixed axis, a plane-parallel or a translational one, and it is driven by means of an inlermcdiary kinematics chain (F

19、igs. 3-5).Further possibilities are shown in Refs. 2-7.X2 /XFig. 3 shows a kinematics schema in which link 2 is joined with the frame at point C, and it is driven by means of the connecting rod 3 from the robot arm 1. The balancing of the forces system that acts on the arm 1 is expressed by the foll

20、owing equation:fi = (/hi OG cos ipj +)g + Fs( Ya cos 0i - XA sin 0) + Ryx Ye -心 ye = 0,i = 1,. ,12,少- J” BGwhere:仿=arctan=丽 /山； 77/44；-Vl£ yEXbYbXcycBCcos 慎sin 慎 sin if/j cos 眞1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-12981297The components of the reaction force be

21、tween the connecting rod 3 and the arm 1, on the axes of fixed co-ordinate system, are:八心一壮）+川3（心一*6）（忌一池）&Yd（Xc 忌）Yc（d 怠）-丫£（肮-尺31人（丫£ 一 Y"） - 73（肮3 一 *D）g X。 Xewhere:T = Fs(Xb 一 XcJsin 0, (沧一人)cos 切 + 叫险 一 Xc) + 心(*6 辰)+ 血(心-Xc)XdYdYcX2D yio1=肉卜惓l+R屣I；R# =blCOS Qi sin J+氏.V3Gj-s

22、in Ci- cos ©The value of anglet/VC/2 + V1- H/2 一 VWibf = arctan,-r7c/2 + v1-w1- vwrepresents the solution of the equation:U cos(心 + a) + K sin(i/rz 4- a) + FT = 0, where:U = 2CD(Xc - XE) V = 2CD( YE- Kc)；W = OE2 + CD2 + OC2 一 DE2 一 1(XEXC + Ua = arctan xidSimilar to the previous case, the angle

23、 of the connecting rod 3 is:CD cos(切 + a) 4- Yc - Xe® = arccosThe distances OG and BG, and the co-ordinates: ”2G 竝八、丫 give the positions of the mass centres of links 1, 4, 3 and 2, respectively.The unknowns of the problem: xm, jt小 xe. yE. .v：/；, Xc. Ya ED. BC、Fo and k are found by solving the s

24、ystem made up through reiterated writing of the equilibrium equation (2) for 12 distinct values of the position angle(p( of the robot arm L which are contained in the work field. The masses j =of the elements and the positions of the mass centresare assumed as known. The static equilibrium of the ro

25、bot arm is accurately realised in those 12 positions according to angles (ph i = 1,. J2 only. Due to continuity reasons, the unbalancing value is negligible between these positions.In fact, the problem is solved in an iterative manner, because at the beginning of the design, the masses of the helica

26、l spring and links 2 and 3 are unknown.The maximum magnitude of the unbalanced moment is inverse proportional to the number of unknowns of the balancing syslem. By assembling the two helical springs in parallel betweenarm 1 and link 25 the balancing accuracy is increased, since 18 distinct values of

27、 angle ® may be imposed within the same work fieldIn Fig. 4, another possibility for the static balancing of a link that rotates around a horizontal fixed axis is shown. The point B belongs to slide 2 which slides along a fixed straight line and is driven by means of the connecting rod 3 by the

28、 robot arm 1. The system, formed by following equilibrium equations:fi = OG cos (fl： + 也X/)g + F$( Ya cos 0 XA sin 0) 冷 + 0, / = 1，11,where(/?-> + my +sin a 代cos(0 a)Z)£* sin a&弭=： Z)Ecos(a-)：C°S 你R|3y =niygDGy cos a cos % (/?h + ?3 + ZbW sin a Fscos(0 a)DE sin 眞Z)Ecos(a W)DE.Xe sin

29、 a Kf cos 7. h e 彷=a + arcsinXn e sin a + (S, + r/)cos a; Y= (Sj + J)sin a e cos a.are solved with respect to the unknowns: xa. va> "id, vid, CD, d. /人 e. a, R)and k. The displacement Sj of the slider has the value:Xe + DE cos 仏一(b + e)sin a，兀cos a,iFif aO.Ye + DE sin i/j + (方 + e)cos a$ =:s

30、in aIf the work field is symmetrical with respect to the vertical axis OY. the balancing mechanism has a particular shape, characterised by yA = yj)= b = e = 0、and a =兀/2 5.The number of the unknowns decreased to six, but the balancing accuracy is higher, because it is possible to consider that the

31、position angles <pf verity the equality:©+6 = x _ 申卜 / = U. ,6.Likewise, the balancing helical spring 4 can be joined to the connecting rod 3 at point B (Fig. 5). Eq. (3) where the components of the reaction force between the arm 1 and link 3 are:(/«2 + 73 + sin a + 代cos(0 a)cos 如 也=cos

32、(i)i 加3(肮3 - 辿 + ng% 一 + 代(巾 一 XJsin 0 -( Y一 )cos 0. DE cos(a i/j)Sm 龟1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-1298#（m2 + ?3 + 74）g sin a + Fscos（0 a）sin i力cos（a 一 心）H（Xg3 _ X。） + n】4B（Xb _ X°）g + 代（X _ X»）sin 0 _ （_ Y»）cos 0DE cos（a 一屮）C°S aXdYd Xe

33、 sin 2 Ye cos a e1力=久 + arcsin -DE1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-1298#1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-12981299is solved with respect to the unknowns: xA, yA. xd, ”小丫3从 CD, e, a, F（）and k.Fig. 6 shows another variant for

34、 the balancing system. The B end of the helical spring 4 is joined to the connecting rod 3 which has a plane-parallel movement. The following unknowns: .V|j, yA> yE.兀3庆）3庆 Xc、Yc> d,斤）and k are found as solutions of the system made up of cquilibiium cqualion （3）, where;悩=5严7);血=心-巴5%and:V = Fs（

35、Xb 一 Xc ）sin 0-（Yti- Yc）cos 0 + “（花 一肮）+ （XGi 一 Xc） +y =人cos（i力 一 0） + m3g sin 如;Fig. 6. Balancing elastic system with oscillating-slidcr mechanism.W = (Yq - K£)sin 切 + (Xq 一 Xe)cos 归;arctang-g-arcsinCE = y/(Xc - -V£)2+( Yc - Ye)2.In the same manner as the constructive solution shown in Fi

36、g. 4, the balancing accuracy is higher, if the work field is symmetrical with respect to the vertical OY axis (yA = jit = V3B = d= Xc = 0) 5, because the position angles(pt verify the equality (4).Fig. 7. Balancing elastic systems for vertical and horizontal robot arms.1. Simionescu, L. Ciupitu ； Me

37、chanism and Machine Theory 35 (20(H) 1287-1298#3. The static balancing of the weight forces of four bar linkage elementsThe static balancing of a vertical arm of a robot presents some particularities, considering that it bears the horizontal arm. For this reason, most of the robot manufacturers use

38、a parallelogram mechanism as a vertical arm (Fig 7). Therefore, the link 3 has a circular translational movement. At point K is joined the elastic system that is used for balancing the weight of the horizontal robot “rm. For balancing of the weight forces of the four-bar linkage elements, any one of

39、 the conslruclive solutions mentioned above can be used. For example, the elastic system schematised in Fig. 3 is considered. The unknown dimensions of the elastic system are found by simultaneously solving the following equations: df©，.,dY© , dYG5 t dYCb加2-Jj- + (加3 + 加8 + 加9 + 川10 + 加11+

40、 "一- + 山5市-+ 叫一-+臥晋+学)H喘=0(5)which are written for 12 distinct values of the position angle(p2i of the vertical arm.These equations result from applying on the virtual power principle to force system which acts on the linkage. The equality (5) is valid when the horizontal arm does not rotate ar

41、ound the axis of pair C、and consequently the velocity of the gravity centre of the ensemble formed by the elements 3, & 9, 10 and 11 is equal to the velocity of point C. The masses of the links and the positions of the gravity centres are supposed to be known.Eq. (5) may be substituted by Eq. (6

42、), if it is assumed that d(p2/f = 1: d%丄,丄丄丄丄、d沧丄dh；dYCb加2- + (也3 + 叫 + mg + ZHjo +| )F + ffl5- + 1)1(.d(p2d(/)2 d(p2Cl<p2d<p21. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-1298#1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-1298#where:-X/)2+(n

43、- Yjf - /O )k;1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-1298#1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-12981301Yg2 = x2g2 sin(p2i + Xig2 cos 他;Yg4 = V464 sin(p2i + y4G4 cos(p2/;Yg. = Yb + X5g5 sin(p5i + 烛 cos(p5/;Yg6 = Yh + X6G6 sin(p6i + V

44、6G6 cos(p&h = Yh + -v6/ sin(pbi + yb/ cos(pbi;Yj = X2J sin cp2j + X2J cos(p2bXf = -Y2A COS(p2j 一 V2F sin(p2bYe = sin(p2j + y1F cos(p2i;Yc = BC sin(p2bvwuju- + r2 - w-g = arctaii :UW-V2- IV2V = 2FG(Xf - XhY r = 2FG( YF - YH)； IV=GHExample - FGA robot arm of mass m = 10 kg is statically balanced w

45、ith the elastic system schematised in Fig. 3, having the following dimensions: DE = 0.100706 m, BC = 0.161528 m, xie = 0.145569 - (Xh-Yf- rz/)2;ST-Rs/R2 + S2-T2=arctan=RT-Ss/R2 + S2 - T2R = 2GH(Xh 一 S S = 2GH( YH 一 ");T = FG2-GH2-(Xb- Xh)2-( Yb - YH)2.The unknowns of the problem are: the length

46、s FG and GH; the co-ordinates: X2F、yib xi/, yu. Xh、Yh、Z、yei of the points F、J、H and J, respectively;the force corresponding to the initial length /(), and the stiffness coeflicient k of the helical spring 7.1. Simionescu, L. Ciupitu ； Mechanism and Machine Theory 35 (20(H) 1287-1298#gravity centre G is OG = 1.0 m. The characteristics of the spring are: the initial l