噴漆機(jī)器人設(shè)計(jì)-大臂系統(tǒng)設(shè)計(jì),噴漆機(jī)器人設(shè)計(jì)-大臂系統(tǒng)設(shè)計(jì),噴漆,機(jī)器人,設(shè)計(jì),系統(tǒng) The static balancing of the industrial robot arms Part I: Discrete balancing Ion Simionescu*, Liviu Ciupitu Mechanical Engineering Department, POLITEHNICA University of Bucharest, Splaiul Independentei 313, RO-77206, Bucharest 6, Romania Received 2 October 1998; accepted 19 May 1999 Abstract The 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 weight forces of the vertical and horizontal arms, many alternatives are shown. Finally, the results of solving a numerical example are presented. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Industrial robot; Static balancing; Discrete balancing 1. Introduction The mechanisms of manipulators and industrial robots constitute a special category of mechanical systems, 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-back programmable robots, where the human operator must drive easily the mechanical system during the training period. Generally, 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 taken Mechanism and Machine Theory 35 (2000) 12871298 0094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S0094-114X(99)00067-1 *Corresponding author. E-mail address: simionform.resist.pub.ro (I. Simionescu). 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 analysed. This balancing 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 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 axis There are several possibilities of balancing the weight forces of the manipulator and robot arms by means of the helical spring elastic forces. The simple solutions are not always applicable. Sometimes an approximate solution is preferred, leading to a convenient alternative from constructional point of view. The simplest balancing possibility of the weight force of a link 1 (the horizontal robot arm, for example) which rotates around 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 to the link 1,is m 1 OG 1 cos j i m 2A X A g F s a 0, i 1,.,6, 1 where the elastic force of the helical spring is: F s F 0 k AB l 0 , and Fig. 1 I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 128712981288 a X B Y A X A Y B AB ; C13 C13 C13 C13 X A Y A C13 C13 C13 C13 R j i C13 C13 C13 C13 x 1A y 1A C13 C13 C13 C13 ; R j i C13 C13 C13 C13 cos j i sin j i sin j i cos j i C13 C13 C13 C13 ; AB X A X B 2 Y A Y B 2 q ; m 2A BG 2 AB m 2 : The gravity centre G 2 of spring 2 is collinear with pairs centres A and B. The stiC128ness coe cient of the spring is denoted by k, m 1 is the mass of the link 1, m 2 is the mass of the helical spring 2, and g represents the gravity acceleration magnitude. Thus, the unknown factors: x 1A , y 1A , X B , Y B , F 0 and k may be calculated in such a way that the equilibrium of the forces is obtained for six distinct values of the angle j i : The movable co- ordinate axis system x 1 Oy 1 attached to the arm 1 was chosen so that the gravity centre G 1 is upon the Ox 1 axis. The co-ordinates x 1A and y 1A defined the position of point A of the arm 1. In the particular case, characterised by y 1A X B l 0 F 0 0, the problem allows an infinite number of solutions, which verify the equation: k m 1 OG 1 m 2A x 1A g x 1A Y B , for any value of angle j: Since in this case, F s kAB(see line 1, Fig. 2), some di culties arise in the construction of this system where it is not possible to use a helical extension spring. The compression spring, which has 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 di cult. Even in the general case, when y 1A 6 0 and X B 6 0, results a reduced value of the initial length l 0 of the spring, corresponding to the forces F 0 0: 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 l 0 from the constructional point of view, may be achieved by replacing the fixed point B of spring articulation by a movable one. In other words, the spring will be articulated with its B end of a movable link 2, whose position depends on that of the 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 intermediary kinematics chain (Figs. 35). Further possibilities are shown in Refs. 27. Fig. 2 I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 12871298 1289 Fig. 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 following equation: f i m 1 OG 1 cos j i m 4A X A g F s Y A cos y i X A sin y i R 31X Y E R 31Y X E 0, i 1,.,12, 2 where: y i arctan Y B Y A X B X A ; m 4A BG 4 AB m 4 ; m 4B m 4 m 4A ; C13 C13 C13 C13 X E Y E C13 C13 C13 C13 R j i C13 C13 C13 C13 x 1E y 1E C13 C13 C13 C13 ; C13 C13 C13 C13 X B Y B C13 C13 C13 C13 C13 C13 C13 C13 X C Y C C13 C13 C13 C13 R c i C13 C13 C13 C13 BC 0 C13 C13 C13 C13 ; R c i C13 C13 C13 C13 cos c i sin c i sin c i cos c i C13 C13 C13 C13 : The components of the reaction force between the connecting rod 3 and the arm 1, on the axes of fixed co-ordinate system, are: R 31X T X D X E m 3 X D X G 3 X C X E g Y D X C X E Y C X D X E Y E X C X D ; R 31Y R 31X Y E Y D m 3 X G 3 X D g X D X E , where: T F s X B X C sin y i Y B Y C cos y i h m 2 X G 2 X C m 3 X G 3 X C m 4B X B X C i g, Fig. 3. Balancing elastic system with four bar mechanism. I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 128712981290 C13 C13 C13 C13 X D Y D C13 C13 C13 C13 C13 C13 C13 C13 X C Y C C13 C13 C13 C13 R c i C13 C13 C13 C13 x 2D y 2D C13 C13 C13 C13 ; C13 C13 C13 C13 X G 2 Y G 2 C13 C13 C13 C13 C13 C13 C13 C13 X C Y C C13 C13 C13 C13 R c i C13 C13 C13 C13 x 2G 2 y 2G 2 C13 C13 C13 C13 ; C13 C13 C13 C13 X G 3 Y G 3 C13 C13 C13 C13 C13 C13 C13 C13 X C Y C C13 C13 C13 C13 R x i C13 C13 C13 C13 x 3G 3 y 3G 3 C13 C13 C13 C13 , R x i C13 C13 C13 C13 cos x i sin x i sin x i cos x i C13 C13 C13 C13 : The value of angle c i : c i arctan U U 2 V 2 W 2 p VW V U 2 V 2 W 2 p UW a represents the solution of the equation: U cos c i a V sin c i a W 0, where: U 2CD X C X E ; V 2CD Y E Y C ; W OE 2 CD 2 OC 2 DE 2 2 X E X C Y E Y C ; a arctan y 2D x 2D : Similar to the previous case, the angle of the connecting rod 3 is: x i arccos CD cos c i a X C X E DE The distances OG 1 and BG 4 , and the co-ordinates: x 2G 2 , y 2G 2 , X G 3 , Y G 3 give the positions of the mass centres of links 1, 4, 3 and 2, respectively. The unknowns of the problem: x 1A , y 1A , x 1E , y 1E , x 2D , y 2D , X C , Y C , ED, BC, F 0 and k are found by solving the system made up through reiterated writing of the equilibrium equation (2) for 12 distinct values of the position angle j i of the robot arm 1, which are contained in the work field. The masses m j , j 1,.,4, of the elements and the positions of the mass centres are assumed as known. The static equilibrium of the robot arm is accurately realised in those 12 positions according to angles j i , i 1,.,12 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 helical 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 system. By assembling the two helical springs in parallel between I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 12871298 1291 arm 1 and link 2, the balancing accuracy is increased, since 18 distinct values of angle j i may be imposed within the same work field. In 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 robot arm 1. The system, formed by following equilibrium equations: f i m 1 OG 1 cos j i m 4A X A g F s Y A cos y X A sin y R 13X Y E R 13Y X E 0, i 1,.,11, 3 where R 13X m 2 m 3 m 4B g sin a F s cos y a DE m 3 gDG 3 sin a DE cos a c i cos c i ; R 13Y m 3 gDG 3 cos a cos c i m 2 m 3 m 4B g sin a F s cos y a DE sin c i DE cos a c i ; c i a arcsin X E sin a Y E cos a b e DE ; X B e sin a S i d cos a; Y B S i d sin a e cos a, are solved with respect to the unknowns: x 1A , y 1A , x 1D , y 1D , CD, d, b, e, a, F 0 and k. The displacement S i of the slider has the value: Fig. 4. Elastic system with slider-crank mechanism I. I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 128712981292 S i X E DE cos c i b e sin a cos a , if a6 p 2 , or S i Y E DE sin c i b e cos a sin a , if a6 0: If the work field is symmetrical with respect to the vertical axis OY, the balancing mechanism has a particular shape, characterised by y 1A y 1D b e 0, and a p=2 5. The number of the unknowns decreased to six, but the balancing accuracy is higher, because it is possible to consider that the position angles j i verify the equality: j i 6 p j i , i 1,.,6: 4 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: R 13X m 2 m 3 m 4B g sin a F s cos y a cos c i cos a c i m 3 X G3 X D m 4B X B X D g F s X B X D sin y Y B Y D cos y DE cos a c i sin a; Fig. 5. Elastic system with slider-crank mechanism II. I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 12871298 1293 R 13Y m 2 m 3 m 4B g sin a F s cos y a sin c i cos a c i m 3 X G3 X D m 4B X B X D g F s X B X D sin y Y B Y D cos y DE cos a c i cos a; C13 C13 C13 C13 X B Y B C13 C13 C13 C13 C13 C13 C13 C13 X D Y D C13 C13 C13 C13 R c i C13 C13 C13 C13 x 3B y 3B C13 C13 C13 C13 , c i a arcsin X E sin a Y E cos a e DE , is solved with respect to the unknowns: x 1A , y 1A , x 1D , y 1D , x 3B , y 3B , CD, e, a, F 0 and k. Fig. 6 shows another variant for 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: x 1A , y 1A , x 1E , y 1E , x 3B , y 3B , X C , Y C , d, F 0 and k are found as solutions of the system made up of equilibrium equation (3), where: R 13X U sin c i V X E X C W ; R 13Y V Y C Y E U cos c i W ; and: U F s X B X C sin y Y B Y C cos y h m 2 X G 2 X C m 3 X G 3 X C m 4B X B X C i g; V F s cos c i y m 3 g sin c i ; Fig. 6. Balancing elastic system with oscillating-slider mechanism. I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 128712981294 W Y C Y E sin c i X C X E cos c i ; c i arctan Y C Y E X C X E arcsin d CE ; CE X C X E 2 Y C Y E 2 q : In the same manner as the constructive solution shown in Fig. 4, the balancing accuracy is higher, if the work field is symmetrical with respect to the vertical OY axis y 1A y 1E y 3B d X C 0 5, because the position angles j i verify the equality (4). Fig. 7. Balancing elastic systems for vertical and horizontal robot arms. I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 12871298 1295 3. The static balancing of the weight forces of four bar linkage elements The 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 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 arm. For balancing of the weight forces of the four-bar linkage elements, any one of the constructive 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: m 2 dY G 2 dt m 3 m 8 m 9 m 10 m 11 dY C dt m 4 dY G 4 dt m 5 dY G 5 dt m 6 dY G 6 dt m 7 2 dY I dt dY J dt g F s dIJ dt 0, 5 which are written for 12 distinct values of the position angle j 2i 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 around the axis of pair C, and consequently the velocity of the gravity centre of the ensemble formed by the elements 3, 8, 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), if it is assumed that dj 2 =dt 1: m 2 dY G 2 dj 2 m 3 m 8 m 9 m 10 m 11 dY C dj 2 m 4 dY G 4 dj 2 m 5 dY G 5 dj 2 m 6 dY G 6 dj 2 m 7 2 dY I dj 2 dY J dj 2 g F s dIJ dj 2 0, 6 where: F s F 0 X I X J 2 Y I Y J 2 q l 0 k; Y G 2 x 2G 2 sin j 2i y 2G 2 cos j 2i ; Y G 4 x 4G 4 sin j 2i y 4G 4 cos j 2i ; Y G 5 Y F x 5G 5 sin j 5i y 5G 5 cos j 5i ; I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 128712981296 Y G 6 Y H x 6G 6 sin j 6i y 6G 6 cos j 6i ; Y I Y H x 6I sin j 6i y 6I cos j 6i ; Y J x 2J sin j 2i y 2J cos j 2i ; X F x 2F cos j 2i y 2F sin j 2i ; Y F x 2F sin j 2i y 2F cos j 2i ; Y C BC sin j 2i ; j 5i arctan VW U U 2 V 2 W 2 p UW V U 2 V 2 W 2 p ; U 2FG X F X H ; V 2FG Y F Y H ; W GH 2 FG 2 X F X H 2 Y F Y H 2 ; j 6i arctan ST R R 2 S 2 T 2 p RT S R 2 S 2 T 2 p ; R 2GH X H X F ; S 2GH Y H Y F ; T FG 2 GH 2 X F X H 2 Y F Y H 2 : The unknowns of the problem are: . the lengths FG and GH; . the co-ordinates: x 2F , y 2F , x 2J , y 2J , X H , Y H , x 6I , y 6I of the points F, J, H and J, respectively; . the force F 0 , corresponding to the initial length l 0 , and the stiC128ness coe cient k of the helical spring 7. 4. Example A robot arm of mass m 1 10 kg is statically balanced with the elastic system schematised in Fig. 3, having the following dimensions: DE 0:100706 m, BC = 0.161528 m, x 1E 0:145569 m, y 1E 0:848205 10 6 m, X C 0:244535 10 3 m, Y C 0:0969134 m, x 1A 0:820178 m, y 1A 0:144475 10 3 m, x 2D 0:0197607 m, y 2D 0:146229 m. The distance to the I. Simionescu, L. Ciupitu/Mechanism and Machine Theory 35 (2000) 12871298 1297 gravity centre G 1 is OG 1 1:0 m. The characteristics of the spring are: the initial length l 0 0:5 m, the stiC128ness coe cient k 3079:38 N/m, and the mass m 4 1:5 kg. In the work field defined by j min 0:785398 and j max 0:785396, the maximum unbalanced moment has the magnitude UM max 0:271177 Nm. References 1 P. Appell, Traite de mecanique rationnelle, Gauthier Villars, Paris, 1928. 2 A. Gopaswamy, P. Gupta, M. Vidyasagar, A new parallelogram linkage configuration for gravity compensation using torsional springs, in: Proceedings of IEEE International Conference on Robotics and Automation, vol. 1, Nice, France, 1992, pp. 664669. 3 K. Hain, Spring mechanisms point balancing, in: N.D. Chironis (Ed.), Spring Design and Application, McGraw-Hill, New York, 1961, pp. 268275. 4 E.P. Popov, A.N. Korenbiashev, Robot Systems, Mashinostroienie, Moscow, 1989. 5 I. Simionescu, L. Ciupitu, On the static balancing of the industrial robots, in: Proceeding of the 4th International Workshop on Robotics in AlpeAdria Region RAA 95, July 68, Po rtschach, Austria, vol. II, 1995, pp. 217220. 6 I. Simionescu, L. Ciupitu, The static balancing of the industrial robot arms, in: Ninth World Congress on the Theory of Machines and Mechanisms, Aug. 29Sept. 2, Milan, Italy, vol. 3, 1995, pp. 17041707. 7 D.A. Streit, E. Shin, Journal of Mechanical Design 115 (1993) 604611. I. Simionescu, L. 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