垂直多關(guān)節(jié)機(jī)器人位置控制結(jié)構(gòu)設(shè)計(jì)（腰部關(guān)節(jié)）

垂直多關(guān)節(jié)機(jī)器人位置控制結(jié)構(gòu)設(shè)計(jì)（腰部關(guān)節(jié)）,垂直,關(guān)節(jié),機(jī)器人,位置,控制,結(jié)構(gòu)設(shè)計(jì),腰部 The static balancing of the industrial robot armsPart I: Discrete balancingIon Simionescu*, Liviu CiupituMechanical Engineering Department, POLITEHNICA University of Bucharest, Splaiul Independentei 313, 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 theindustrial robot arms, using the elastic forces of the helical springs. For the balancing of the weightforces of the vertical and horizontal arms, many alternatives are shown. Finally, the results of solving anumerical example are presented. 7 2000 Elsevier Science Ltd. All rights reserved.Keywords: Industrial robot; Static balancing; Discrete balancing1. IntroductionThe mechanisms of manipulators and industrial robots constitute a special category ofmechanical systems, characterised by big mass elements that move in a vertical plane, withrelatively slow speeds. For this reason the weight forces have a high share in the category ofresistance that the driving system must overcome. The problem of balancing the weight forcesis extremely important for the play-back programmable robots, where the human operatormust drive easily the mechanical system during the training period.Generally, the balancing of the weight forces of the industrial robot arms results in thedecrease of the driving power. The frictional forces that occur in the bearings are not takenMechanism and Machine Theory 35 (2000) 128712980094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.PII: S0094-114X(99)00067- Corresponding author.E-mail address: simionform.resist.pub.ro (I. Simionescu).into consideration because the frictional moment senses depend on the relative movementsenses.In this work, some possibilities of balancing of the weight forces by the elastic forces of thecylindrical helical springs with straight characteristics are analysed.This balancing can be made discretely, for a finite number of work field positions, or incontinuous mode for all positions throughout the work field. Consequently, the discretesystems realised only an approximatively balancing of the arm.The use of counterweights is not considered since they involve the increase of movingmasses, 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 robotarms by means of the helical spring elastic forces.The simple solutions are not always applicable. Sometimes an approximate solution ispreferred, 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. Ahelical spring 2, joined between a point A of the link and a fixed B one, is used. The equationthat expresses the equilibrium of the forces moments 1, which act to the link 1, is?m1OG1cos ji? m2AXA?g ? Fsa ? 0, i ? 1,.,6,?1?where the elastic force of the helical spring is:Fs? F0? k?AB ? l0?,andFig. 1I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981288a ?XBYA? XAYBAB;?XAYA? Rji?x1Ay1A?;Rji?cos ji?sin jisin jicos ji?;AB ?XA? XB?2?YA? YB?2q;m2A?BG2ABm2:The gravity centre G2of spring 2 is collinear with pairs centres A and B.The sti?ness coe?cient of the spring is denoted by k, m1is the mass of the link 1, m2is themass of the helical spring 2, and g represents the gravity acceleration magnitude.Thus, the unknown factors: x1A, y1A, XB, YB, F0and k may be calculated in such a way thatthe equilibrium of the forces is obtained for six distinct values of the angle ji: The movable co-ordinate axis system x1Oy1attached to the arm 1 was chosen so that the gravity centre G1isupon the Ox1axis. The co-ordinates x1Aand y1Adefined the position of point A of the arm 1.In the particular case, characterised by y1A? XB? l0? F0? 0, the problem allows aninfinite number of solutions, which verify the equation:k ?m1OG1? m2Ax1A?gx1AYB,for any value of angle j:Since in this case, Fs? k AB (see line 1, Fig. 2), some di?culties arise in the construction ofthis 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 moredi?cult.Even in the general case, when y1A6?0 and XB6?0, results a reduced value of the initial lengthl0of the spring, corresponding to the forces F0? 0: The modification of the straightcharacteristic position to the necessary spring for balancing (line 2, Fig. 2), i.e. to obtain anacceptable initial length l0from the constructional point of view, may be achieved by replacingthe fixed point B of spring articulation by a movable one. In other words, the spring will bearticulated 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 translationalone, and it is driven by means of an intermediary kinematics chain (Figs. 35).Further possibilities are shown in Refs. 27.Fig. 2I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981289Fig. 3 shows a kinematics schema in which link 2 is joined with the frame at point C, and itis driven by means of the connecting rod 3 from the robot arm 1. The balancing of the forcessystem that acts on the arm 1 is expressed by the following equation:fi?m1OG1cos ji? m4AXA?g ? Fs?YAcos yi? XAsin yi? ? R31XYE? R31YXE? 0,i ? 1,.,12,?2?where: yi? arctanYB?YAXB?XA; m4A?BG4ABm4; m4B? m4? m4A;?XEYE? Rji?x1Ey1E?;?XBYB?XCYC? Rci?BC0?;Rci?cos ci?sin cisin cicos ci?:The components of the reaction force between the connecting rod 3 and the arm 1, on the axesof fixed co-ordinate system, are:R31X?T?XD? XE? ? m3?XD? XG3?XC? XE?gYD?XC? XE? ? YC?XD? XE? ? YE?XC? XD?;R31Y?R31X?YE? YD? ? m3?XG3? XD?gXD? XE,where:T ? Fs?XB? XC?sin yi? ?YB? YC?cos yi?hm2?XG2? XC? m3?XG3? XC? m4B?XB? XC?ig,Fig. 3. Balancing elastic system with four bar mechanism.I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981290?XDYD?XCYC? Rci?x2Dy2D?;?XG2YG2?XCYC? Rci?x2G2y2G2?;?XG3YG3?XCYC? Rxi?x3G3y3G3?,Rxi?cos xi?sin xisin xicos xi?:The value of angle ci:ci? arctanU?U2? V2? W2p? VW?V?U2? V2? W2p? UW? arepresents the solution of the equation:U cos?ci? a? V sin?ci? a? W ? 0,where:U ? 2CD?XC? XE?;V ? 2CD?YE? YC?;W ? OE2? CD2? OC2? DE2? 2?XEXC? YEYC?;a ? arctany2Dx2D:Similar to the previous case, the angle of the connecting rod 3 is:xi? arccosCD cos?ci? a? XC? XEDEThe distances OG1and BG4, and the co-ordinates: x2G2, y2G2, XG3, YG3give the positions of themass centres of links 1, 4, 3 and 2, respectively.The unknowns of the problem: x1A, y1A, x1E, y1E, x2D, y2D, XC, YC, ED, BC, F0and k arefound by solving the system made up through reiterated writing of the equilibrium equation (2)for 12 distinct values of the position angle jiof the robot arm 1, which are contained in thework field. The masses mj, j ? 1,.,4, of the elements and the positions of the mass centresare assumed as known. The static equilibrium of the robot arm is accurately realised in those12 positions according to angles ji, i ? 1,.,12 only. Due to continuity reasons, theunbalancing 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 numberof unknowns of the balancing system. By assembling the two helical springs in parallel betweenI. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981291arm 1 and link 2, the balancing accuracy is increased, since 18 distinct values of angle jimaybe imposed within the same work field.In Fig. 4, another possibility for the static balancing of a link that rotates around ahorizontal fixed axis is shown. The point B belongs to slide 2 which slides along a fixedstraight line and is driven by means of the connecting rod 3 by the robot arm 1. The system,formed by following equilibrium equations:fi?m1OG1cos ji? m4AXA?g ? Fs?YAcos y ? XAsin y? ? R13XYE? R13YXE? 0,i ? 1,.,11,?3?whereR13X?m2? m3? m4B?g sin a ? Fscos?y ? a?DE ? m3gDG3sin aDE cos?a ? ci?cos ci;R13Y?m3gDG3cos a cos ci?m2? m3? m4B?g sin a ? Fscos?y ? a?DE sin ciDE cos?a ? ci?;ci? a ? arcsinXEsin a ? YEcos a ? b ? eDE;XB? e sin a ? ?Si? d?cos a;YB? ?Si? d?sin a ? e cos a,are solved with respect to the unknowns: x1A, y1A, x1D, y1D, CD, d, b, e, a, F0and k.The displacement Siof the slider has the value:Fig. 4. Elastic system with slider-crank mechanism I.I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981292Si?XE? DE cos ci? ?b ? e?sin acos a,if a6?p2,orSi?YE? DE sin ci? ?b ? e?cos asin a,if a6?0:If the work field is symmetrical with respect to the vertical axis OY, the balancingmechanism has a particular shape, characterised by y1A? y1D? b ? e ? 0, and a ? p=2 5.The number of the unknowns decreased to six, but the balancing accuracy is higher, becauseit is possible to consider that the position angles jiverify the equality:ji?6? p ? ji,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:R13X?m2? m3? m4B?g sin a ? Fscos?y ? a?cos cicos?a ? ci?m3?XG3? XD? ? m4B?XB? XD?g ? Fs?XB? XD?sin y ? ?YB? YD?cos y?DE cos?a ? ci?sin a;Fig. 5. Elastic system with slider-crank mechanism II.I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981293R13Y?m2? m3? m4B?g sin a ? Fscos?y ? a?sin cicos?a ? ci?m3?XG3? XD? ? m4B?XB? XD?g ? Fs?XB? XD?sin y ? ?YB? YD?cos y?DE cos?a ? ci?cos a;?XBYB?XDYD? Rci?x3By3B?,ci? a ? arcsinXEsin a ? YEcos a ? eDE,is solved with respect to the unknowns: x1A, y1A, x1D, y1D, x3B, y3B, CD, e, a, F0and k.Fig. 6 shows another variant for the balancing system. The B end of the helical spring 4 isjoined to the connecting rod 3 which has a plane-parallel movement. The following unknowns:x1A, y1A, x1E, y1E, x3B, y3B, XC, YC, d, F0and k are found as solutions of the system made upof equilibrium equation (3), where:R13X?U sin ci? V?XE? XC?W;R13Y?V?YC? YE? ? U cos ciW;and:U ? Fs?XB? XC?sin y ? ?YB? YC?cos y?hm2?XG2? XC? m3?XG3? XC? m4B?XB? XC?ig;V ? Fscos?ci? y? m3g sin ci;Fig. 6. Balancing elastic system with oscillating-slider mechanism.I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981294W ? ?YC? YE?sin ci? ?XC? XE?cos ci;ci? arctanYC? YEXC? XE? arcsindCE;CE ?XC? XE?2?YC? YE?2q:In the same manner as the constructive solution shown in Fig. 4, the balancing accuracy ishigher, if the work field is symmetrical with respect to the vertical OY axis ?y1A? y1E? y3B?d ? XC? 0? 5, because the position angles jiverify the equality (4).Fig. 7. Balancing elastic systems for vertical and horizontal robot arms.I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 1287129812953. The static balancing of the weight forces of four bar linkage elementsThe static balancing of a vertical arm of a robot presents some particularities, consideringthat it bears the horizontal arm. For this reason, most of the robot manufacturers use aparallelogram mechanism as a vertical arm (Fig. 7). Therefore, the link 3 has a circulartranslational movement. At point K is joined the elastic system that is used for balancing theweight of the horizontal robot arm. For balancing of the weight forces of the four-bar linkageelements, any one of the constructive solutions mentioned above can be used. For example, theelastic system schematised in Fig. 3 is considered. The unknown dimensions of the elasticsystem are found by simultaneously solving the following equations:?m2dYG2dt? ?m3? m8? m9? m10? m11?dYCdt? m4dYG4dt? m5dYG5dt? m6dYG6dt?m72?dYIdt?dYJdt?g ? FsdIJdt? 0,?5?which are written for 12 distinct values of the position angle j2iof the vertical arm.These equations result from applying on the virtual power principle to force system whichacts on the linkage. The equality (5) is valid when the horizontal arm does not rotate aroundthe axis of pair C, and consequently the velocity of the gravity centre of the ensemble formedby the elements 3, 8, 9, 10 and 11 is equal to the velocity of point C. The masses of the linksand the positions of the gravity centres are supposed to be known.Eq. (5) may be substituted by Eq. (6), if it is assumed that dj2=dt ? 1:?m2dYG2dj2? ?m3? m8? m9? m10? m11?dYCdj2? m4dYG4dj2? m5dYG5dj2? m6dYG6dj2?m72?dYIdj2?dYJdj2?g ? FsdIJdj2? 0,?6?where:Fs? F0?XI? XJ?2?YI? YJ?2q? l0?k;YG2? x2G2sin j2i? y2G2cos j2i;YG4? x4G4sin j2i? y4G4cos j2i;YG5? YF? x5G5sin j5i? y5G5cos j5i;I. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981296YG6? YH? x6G6sin j6i? y6G6cos j6i;YI? YH? x6Isin j6i? y6Icos j6i;YJ? x2Jsin j2i? y2Jcos j2i;XF? x2Fcos j2i? y2Fsin j2i;YF? x2Fsin j2i? y2Fcos j2i;YC? BC sin j2i;j5i? arctanVW ? U?U2? V2? W2pUW ? V?U2? V2? W2p;U ? 2FG?XF? XH?;V ? 2FG?YF? YH?;W ? GH2? FG2? ?XF? XH?2?YF? YH?2;j6i? arctanST ? R?R2? S2? T2pRT ? S?R2? S2? T2p;R ? 2GH?XH? XF?;S ? 2GH?YH? YF?;T ? FG2? GH2? ?XF? XH?2?YF? YH?2:The unknowns of the problem are:. the lengths FG and GH;. the co-ordinates: x2F, y2F, x2J, y2J, XH, YH, x6I, y6Iof the points F, J, H and J, respectively;. the force F0, corresponding to the initial length l0, and the sti?ness coe?cient k of the helicalspring 7.4. ExampleA robot arm of mass m1? 10 kg is statically balanced with the elastic system schematised inFig. 3, having the following dimensions: DE ? 0:100706 m, BC = 0.161528 m, x1E? 0:145569m, y1E? ?0:848205 ? 10?6m, XC? 0:244535 ? 10?3m, YC? 0:0969134 m, x1A? 0:820178m, y1A? 0:144475 ? 10?3m, x2D? 0:0197607 m, y2D? ?0:146229 m. The distance to theI. Simionescu, L. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981297gravity centre G1is OG1? 1:0 m. The characteristics of the spring are: the initial length l0?0:5 m, the sti?ness coe?cient k ? 3079:38 N/m, and the mass m4? 1:5 kg.In the work field defined by jmin? ?0:785398 and jmax? 0:785396, the maximumunbalanced moment has the magnitude UMmax? 0:271177 Nm.References1 P. Appell, Traite de me canique rationnelle, Gauthier Villars, Paris, 1928.2 A. Gopaswamy, P. Gupta, M. Vidyasagar, A new parallelogram linkage configuration for gravity compensationusing 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 4thInternational 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 theTheory 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. Ciupitu / Mechanism and Machine Theory 35 (2000) 128712981298