【机械类毕业论文中英文对照文献翻译】工业机器人手臂的静态平衡
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机械类毕业论文中英文对照文献翻译
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【机械类毕业论文中英文对照文献翻译】工业机器人手臂的静态平衡,机械类毕业论文中英文对照文献翻译,机械类,毕业论文,中英文,对照,文献,翻译,工业,机器人,手臂,静态,平衡
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工业机器人手臂的静态平衡第一部分：平衡离散Ion Simionescu*, Liviu CiupituMechanical Engineering Department, POLITEHNICA University of Bucharest, Splaiul Independentei 313, RO-77206,Bucharest 6, RomaniaReceived 2 October 1998; accepted 19 May 1999摘要：本文介绍了一些在工业机器人手臂的重量平衡解决方案，运用了螺旋弹簧的弹性力量。 垂直和水平手臂的重量力量的平衡显示很多备选方案。 最后，举例子，解决一个数值示例。关键词：工业机器人；静态平衡；离散平衡7 2000 Elsevier Science Ltd. All rights reserved. 1. 介绍 机器人及工业机器人机制构成了一个特殊类别的机器系统，其特点是大质量的元素在一个垂直平面移动速度相对缓慢。基于这个原因，重量势力成了驱动系统必须要克服的一大份额的阻力。对于平衡重量力量的问题，可编程序的机器人是非常重要的，在训练期间，人工操作必须容易地驾驶机械系统。一般来说，工业机器人手臂的重量平衡力量都将会削弱驱动力量。在轴承发生的摩擦力没有被考虑到，因为摩擦时刻感觉取决于相对运动感觉。在这项工作中，对直圆柱螺旋弹簧弹力影响力量平衡问题的可能性进行了分析。这种平衡的可以被分离出来，可以是工作领域位置的有限数字，或者在在工作领域中的所有位置的连续。 因此，离散系统只能实现了机器人手臂的近似平衡。增量的使用并没有被考虑在内，因为他们涉及到了移动的质量物体的增加，整体大小，惯性和组分的压力。2. 在一固定水平轴附近的重量力量的平衡通过螺旋弹簧的弹力来平衡机器手和机器人的重量力量，有集中可行的方案。简单的解决方案并不总是适用的。有时候从建筑角度来首选一个有效的近似解替代原先方案。在一个水平固定轴附近的链接1（例如：横向机械手臂）的重量力量的维持平衡的最简单的方法在图1中该要的显示出来了。在链接点A和固定点B之间，使用了一个螺旋弹簧2.以下是对链接1适用的表达力矩的平衡公式：(m1OG1cosi+m2A)g+Fsa=0,i=1,6在那里，螺旋弹簧弹力是: FS=F+k(AB-l0),和弹簧2的重心G2和双中心A、B两点在同一个直线上。弹簧的弹性系数由 k 表示、 m1 是链接 1 的质量、 m2 是 螺旋弹簧2的质量 ， g 表示重力加速度的大小。这样，通过六个非重复值i以及由其获得的力的平衡值，可以获得以下的未知值：1A,y1A,XB,YB,F0和K 。为了使得重心G1位于OX1 上，对于手臂1我们选择活动协调轴系统X1 OY1 . X1A 和Y1A 的调整确定了臂1上点A的位置。 在一些特殊的情况下，当y1A=XB=l0=F0=0 时，这个问题可以有无限的解答，通过下面的公式定义：k=,角度取任意值。因为在这种情况下， FS=k AB（见图2 第一行），不使用螺旋弹簧的系统在建筑上出现了一些困难。压缩弹簧，它对于计算的功能，不能被对折。因此，在导航中出现的摩擦力使得培训工作更加困难。甚至于在一般的情况下，当y1A0和XB0时，弹簧的初始长度l0 的减少，相当于力F0=0。对于平衡所必须的弹簧的平直特征位置的径向变位系数（图2直线2），换言之，从建筑学的角度上看，为了获得一个可以接受的原始长度l0 ,可能可以用一个移动的弹簧取代固定B点的弹簧连接。换句话来说，弹簧的B端挂在可移动的链接2上，位置随着手臂1的变化而变化。链接2可能有一个平面副的或者是直线的绕着一个固定点的转动运动副，并且它通过中介动力学链子所驱动。（图3-5）在引用里展示了更多的可能性2-7。 图3. 弹性系统的平衡与四杆机构图3展示了一个运动学构架，其中连接2在C点帧加入，它通过连接杆3和机器人手臂1的链接进行驱动。在手臂1运行的平衡力量系统由一下方程表示：fi=(m1OG1cos+m4AXA)g+Fs(YAcosXAsin)+R31XYER31YXE=0,i=1,，12， （2）在连接杆3和机器人手臂1之间的反作用力组分，在固定坐标系轴上：类似于前面的例子，连接杆3的角度是：OG1 和BG4的距离，同,分别决定了链接1、4、2.2 的质量重心的位置。未知数 , ,ED, BC, 和k通过解决平衡方程（2）解得，其中需要工作区域12个机器人手臂的非重复位置角i 。元素的质量mj ( j=1,.,4)和物质中心假设是已知的。根据那些角：i,i=1,，12机器人手臂的静态平衡在那些12个位置保持平衡。由于连续性的原因，不平衡值在这些位置上是微不足道的。 实际上，问题是以一种反复的方式解决的，因为在设计之初，关于螺旋弹簧和链接2和3的情况，很多都是未知的。不平衡力矩的最大值和平衡系统的未知数成反比。通过在臂1和链接2上两个平行圆柱螺旋弹簧的组装，平衡精度增加了，因为18个非重复值的i可施加在相同的工作领域。 在 Fig.4 中，显示了围绕一个固定的横轴的链接的静态平衡的另一种可能性。被固定在直线上滑行的滑道2上的B点通过机器人手臂由杆3驱动。该系统根据以下的平衡方程形成：fi=(m1OG1cos+m4AXA)g+Fs(YAcosXAsin)+R13XYER13YXE=0,i=1,，11， (3)未知数：,CD,d,b,e,a,and k。滑块的位移Si可以取以下的值： 图.5.弹性系统与曲柄滑块机构.如果工作领域关于垂直轴OY对称，那么平衡机制就有一个特定的模式，并由这些变量决定：y1A=y1D=b=e=0,和 5。未知值减少到了六个 ，但是平衡精度提高了，因为考虑到了位置角i决定了以下的方程式：，i=1,6. (4)同样，平衡螺旋弹簧4可以在B点加入到连杆点3.。(Fig.5).Eq.(3) 臂1和链接3之间的反应力的构成为：未知数为：,CD,e,a,and k。 图6显示了另一个平衡系统变体。螺旋弹簧4B端加入了能够平面平行运动的连杆3.以下的未知数,d,和 k.被作为由以下平衡方程构筑的系统的解决方案(3)：和 图.6. 弹性系统的平衡与振荡滑块机构.一样的方法，如果工作领域关于垂直轴Oy对称.(y1A=y1E=y3B=d=XC=0)5的话，在图4显示的建设性的解决方案，平衡精度性更高，因为位置角i决定了方程式。 图.7. 纵向和横向平衡的机器人手臂弹性系统.3、四连杆结构的重力的静态平衡由于机器人垂直壁承载着水平臂的问题，机器人垂直臂的静态平衡显示出了一些特殊情况。基于这个原因，大多数的机器人制造商选择使用平行四边形模型作为一个垂直臂。(如图.7)因此，链接3有一个圆形平移运动。在K点加入了弹性系统，是为了平衡水平机器手臂重量。以上的任何一个方案都可以解决四连杆元素的重量力平衡问题。例如，图3的弹性系统。弹性系统的未知尺寸同时解决了下面的方程：以上这个方程所写的12个垂直臂可变位置角的值。这些方程是虚功原理应用于链接系统的成果。当水平的手臂不旋转绕轴 C，而因此由 3,8,9,10 和 11 几元素组成的重心的速度等于点 C.的速度时，等式（5）是成立的。所有的链接和重心的位置都应该是已知的。等式（5）可以被等式（6）替代，如果d2/dt=1成立：以下是未知值：l FG和GH的长度；l 坐标：点F,J,H 和 J的坐标；,l 对应于原始长度l0 和刚性弹簧系数k 的F04. 举例机器人手臂质量m1=10kg 和 图3的弹性系统处于静态平衡状态，已知：DE =0.100706 m, BC = 0.161528 m, x1E =0.145569m, y1E =0.84820106 m, XC =0.244535103 m, YC = 0.0969134 m, x1A =0.820178m, y1A= 0.144475103 m, x2D=0.0197607 m, y2D= 0.146229 m。重心G1有OG1=1.0m 。关于弹簧有 原始长度l0 =0.5m 弹性系数k=3079.38N/m ,弹簧重m4 =1.5 kg 。当min=0.785398和max=0.785396时，最大不平衡时刻有最大值，最大值UMmax=0.271177 Nm。参考文献： 1 P. Appell, Traite de mecanique rationnelle, Gauthier Villars, Paris, 1928.2 A. Gopaswamy, P. Gupta, M. Vidyasagar, A new parallelogram linkage conguration 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.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)00067-1/locate/mechmt* 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
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