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南京理工大学泰州科技学院毕业设计(论文)外文资料翻译系部： 机械工程系 专 业： 机械工程及自动化 姓 名： 尚征瑞 学 号： 05010133 外文出处： Computers & Structures Vol.65, No.2,pp255-259,1997 Elsevier Science 附 件： 1.外文资料翻译译文；2.外文原文。 指导教师评语： 签名： 年 月 日注：请将该封面与附件装订成册。附件1：外文资料翻译译文具有动态特性约束的高速灵活的机械手优化设计摘要：本文提出了一种强调时间独立和位移约束的机器手优化设计理论，该理论用数学编程的方法给予了实现。将各元件用灵活的连杆连接起来。设计变量即为零件横截面尺寸。另用最关键的约束等量替换时间约束。结果表明，此方法产生的设计结果比运用Kresselmeier-Steinhauser函数，且利用等量约束所产生的设计方案更好。建立了序列二次方程基础上的优化设计方案，且设计灵敏度通过总体有限偏差来评定。动态非线性方程组包含了有效运动和实际运动的自由度。为了举例说明程序，设计了一款平面机器人，其中利用某一特定的方案并且运用了不同的等量约束进行了设计。 版权属于 1997年埃尔塞维尔科技有限公司1 导论目前对高速机器人的设计要求越来越高，元件质量的最小化是必不可少的要求。传统机器手的设计取决于静态体系中运动方式的多样化，但这并不适合于高速系统即应力和绕度均受动力效应控制的系统。为了防止失败，在设计的时候必须考虑到有效轨迹和实际运动轨迹之间的相互影响。在暂态负载下对结构系统进行设计已经开始展开研究，该研究是基于下面几个不同的等量约束条件下进行的，分别为对临界点的选择上1 ， 反约束的时间限制2 ，和Kreisselmeier - Steinhauser函数3,4的基础上进行研究。在选择临界点时，假定临界点的位置的时间是固定的，然而这种假设不适合高速系统。第二个办法的缺点是等量约束在可行域内几乎为0，因此现在还没有迹象表明这些约束是否重要。使用Kreisselmeier - Steinhauser函数在可行域中产生了非零的等量约束，但它定义了一个保守的约束，从而产生了一个过于安全的设计方法。 在设计机器手的时候，常规方法是考虑多静态姿态5-7，而不是考虑时间上的约束。这种方法并不适合高速系统，原因是一些姿态不能代表整个系统的运动，此外，位移和应力的计算也是不准确的，这是因为在计算的时候省略了刚性和弹性运动之间的联系。事实上，这种联系是灵活多体分析中最基本的8-10 。 在这项研究中，开发了一种设计高速机械手的方法，这种方法考虑了系统刚性弹性运动之间的联系及时间独立等约束。把最关键的约束作为等量约束。 最关键的约束的时间点可能随着设计变量值的变化而变化。反应灵敏度由整体偏移所决定，设计的最优化取决于序列二次方程式。为了说明程序， 对双杆平面机器手的强度和刚度进行了优化。设计结果与那些采用了Kreisselmeier - Steinhauser函数的机器手进行对比。2、设计理念在这一节中，机器手的优化设计方法使用用于计算强度和刚性的非线性数学编程方法。机器手由N个活动连杆组成，每一个连杆由Ek个有限零件柱组成。其目的是尽可能的减小机械手的质量。与强度关联的约束主要是应力元素和刚性约束。这些约束将使得有效运动的位移产生偏移。设计变量就是连杆和零件的截面特性。从数学上来说，目标函数应满足这样的约束: （1）其中和分别是第k个机构的第i个零件的密度和体积，x是设计变量的矢量，是时间约束总数。在验证位移和应力的时候，参考文献10中的递推公式可用来计算机器手有效轨迹与实际轨迹。将连杆的变形与连杆参照系联系起来，其中在一定边界约束条件下做完整运动。这样通过缩小模型就可以减少每个连杆的实际自由度数了。 系统的广义坐标系是由连杆变量和模块变量组成的。微粒P的运动速度可表式为 (2)其中和是相互制约的系数。凯恩（Kane）等人的方程式12曾被用来测定一些运动方程式如 (3)其中是整体速度向量，F是合成外力向量，M、Q还有分别为总质量、柯氏力、地心引力和弹力，计算公式如下： （4） （5） (6)其中上标r和f分别代表有效自由度和实际自由度。K为对角矩阵，其对角线上的子矩阵是减少了的有效矩阵以连杆变量的形式出现的。为了验证子矩阵在方程（4，5）中是否正确，和可表示如下： p, r=1,2,3; q=1,; s=1, ,12 (7a) p, r=1,2,3; q=1,m; s=1,12 (7b) 其中是元件形状函数，是连杆变量数，m是模块变量数。方程式中的标注即多次出现的下标指数是以概括的形式出现的，这些下标只不过是公式的一部分，并不表示某一含义除非特定指明。这些子矩阵可表示成： 其中和；z,u=1,2,3； s,v=1,12是时间变量，是第k个机构的第i个元件的质量。在定义和时，柯氏力和地心引力可由下列算式计算出来： 这个运动方程式综合了变量步长和变量预测校正的算法，以获取坐标系和中的时间记录。于是，有关物体参考系的节点位移可由模块转换公式获得。由应力与位移关系式计算出零件受到的压应力。整个参考系中各点的位移可用和机架的各节点位移算出。点的偏移可由那个点在实际运动和有效运动的位移差精确的求出。应当指出的是，在运动方程式中，设计变量函数的形式有矩阵，零件的质量和初始矢量中的、阵列。因此在对灵敏度进行分析的时候，这些都应与设计变量区分开来。然而，分析并且验证灵敏度在这次研究中是个非常困难的项目。不全面的分析或是允许极小误差的方式来研究这一问题也未尝不是个好方法。3.减少约束对机器手进行动态分析的方法就是计算个独立点在同一时间内的运动。因此，约束数目最好满足 ，而且这么多的约束在优化设计时也是不切实际的。不过有一个很有效的办法可以使约束数控制在范围内又可以使约束数满足t的所有值，这就是用Kreisselmeier - Steinhauser函数 3 等量替换单个时间约束，此函数表示如下： 其中和C是正数并由和之间的关系决定即min().这可以说明Kreisselmeier-Steinhauser函数限定了一个保守的值域4比如总是比min()更重要，而且c的值越大和min()之间的差就越小。这就是所谓用最关键的约束等量替换了诸如 （11）之类的约束。在这一方法中，用等量约束限定了分段函数并使其由向间断的过渡。在这一值域里尽管左右突出的构件在过渡点有差异，但他们具有相同的标识和梯度，因此可在过渡点自然结合。随着时间逐步的趋近零点，等量约束也变得逐渐光滑。上述所提到的非线性约束优化问题可以由NLPQL11来解决，即运用序列二次方程的方法。这种优化需要初始信息和，m=1, 这两个可由目前研究出的有限差来计算。4.举例双杆平面机器人如图1所示。运动原理是被动块E沿直线从初始位置（1=120，2=-150）运动到终点位置（1=60，2=-30）。E的运动轨迹表示如下：整个运动过程的时间T=0.5s。 每一个连杆的长度为0.6米并由两个等长的零件连接着。其零件的外径，其为本设计的变量，k=1,2；i=1,2。零件的厚度为0.1。物体的压强和密度分别是E=72GPa,=2700Kg/m-3。模块变量缩小了形状尺寸。最先结合的两个模块和最先有着固定自由的约束条件的轴也都被考虑到了。位于连接点B处的杆2质量为2kg，被动物块和有效载荷的总质量为1kg。设计的约束条件如下：-75MPai75MPa i=1, 0,001m其中应力约束由节点顶部或底部的个点来验证。是E的实际运动轨迹与有效运动轨迹的偏离量（即x和y方向的最大偏移值）。初始设计变量均为50mm. 图1 平面机器手操作器在这个例子里，等量约束是由最关键的约束组成的并且其结果与Kreisselmeier-Steunhauser函数的结果进行了比较。后者函数中适用了c的不同值，可以发现c的值越小其产生的设计就越死板。c=50时的设计是最理想的。应当指出的是编译器的限制可能会超过c的最大值，这完全取决于指数函数也就是只要设计变量的低限足够的小。另一方面，最关键的约束会产生极小质量的设计并且精确的迎合偏移位移量。最小的质量，恰当的直径和反复运动的次数在表1中列出。设计轨迹见表2。表KS-c表明了由Kreisselmeier-Steinhauser函数产生的结果，然而MCC表示关键约束。可见应力远远小于允许值，因此应力约束受到了限制。连杆2中间的应力最大（见）图3。被动物块的偏移量的最佳解决方案见图4图2 设计参数表1 平面机器人控制器最佳方法图3 顶部连接两个的平均压力的最佳设计图4 最终效应器偏差的最佳设计5.总结在研究中，高速遥控操纵器的最佳设计方案取决于动态特性。操纵器的固定轨迹与实际轨迹运动也必须考虑到。把最关键的约束用作等量约束。 最关键的约束的时间点可能随着设计变量的改变而变化。这表明分段的等量约束并不会使设计过程产生缺陷。序列二次方程用于解决设计问题，其是运用整体偏差进行灵敏度计算。 高速平面遥控操纵器已被优化设计成在应力和偏差限制下的最小质量。基于Kreisselmeier - Steinhauser函数产生的保守设计下使用等量约束，最好的设计理念就是用最关键的约束。附件2：外文原文（复印件） 南京理工大学泰州科技学院毕业设计说明书(论文)作 者:尚征瑞学 号：05010133系部:机械专 业:机械工程及自动化题 目:送料机械手的设计讲师曹春平指导者： 张卫高级工程师评阅者： 2009 年 5 月Compurrrs & Strucrures Vol. 65. No. 2, pp. 255-259, 1997 0 1997 Elsevier Science Ltd. All rights reserved Pergamon PII: SOO45-7949(96)00269-6 Printed in &eat Britain Gu45-7949/97 Sl7.00 + 0.00 OPTIMUM DESIGN OF HIGH-SPEED FLEXIBLE ROBOTIC ARMS WITH DYNAMIC BEHAVIOR CONSTRAINTS S. Oral and S. Kemal Ider Department of Mechanical Engineering, METU, Ankara 06531, Turkey (Received 31 May 1995) Abstract-A methodology is presented for the optimum design of robotic arms under time-dependent stress and displacement constraints by using mathematical programming. Finite elements are used in the modeling of the flexible links. The design variables are the cross-sectional dimensions of the elements. The time depenclence of the constraints is removed through the use of equivalent constraints based on the most critical constraints. It is shown that this approach yields a better design than using equivalent constraints obtained by the Kresselmeier-Steinhauser function. An optimizer based on sequential quadratic programming is used and the design sensitivities are evaluated by overall finite differences. The dynamical equations contain the nonlinear interactions between the rigid and elastic degrees-of-freedom. To illustrate the procedure, a planar robotic arm is optimized for a particular deployment motion by using different equivalent constraints. 0 1997 Elsevier Science Ltd. 1. INTRODUCTION The increasing demand for high-speed robots has made it necessary to use components that must be designed for minimum weight. The traditional design of robotic arms based on multiple postures in static regime is not suitable for high-speed systems where the stresses and deflections are governed by the dynamic effects. To prevent failure, intricate inter- actions between the rigid and elastic motions must be taken into account in the design. The design of structural systems under transient loading has been studied by using different equivalent constraints based on critical point selection 11, time integral of violated constraints 2, and Kreis- selmeier-Steinhauser function 3,4. In critical point selection, it is assumed that the location of the critical points are assumed to be fixed in time, however this assumption is not appropriate for high-speed multibody systems The second approach has the disadvantage that the equivalent constraint is zero in the feasible domain and hence there is no indication when the constraint is almost critical. The use of Kreisselmeier-Steinhauser function results in an equivalent constraint which is nonzero in the feasible domain, however it defines a conservative envelope and yields oversafe designs. In the design of robotic arms, the conventional approach is to consider multiple static postures 5-71 rather than considering the time-dependency of the constraints. This approach is not appropriate for high-speed systems, since a few postures cannot represent the overall system motion, and furthermore the displacements and stresses computed are inaccur- ate due to omitting the coupling between rigid and elastic motions. In fact, this coupling is the essence of a flexible multibody analysis 8-lo. In this study, a methodology for the design of high-speed robotic arms is developed considering the coupled rigid-elastic motion of the system and the time-dependency of the constraints. The most critical constraints are used as the equivalent constraints. The time points of the most critical constraints may vary as the design variables change. The sensitivity of the response is evaluated by overall finite differences and the optimization is carried out by sequential quadratic programming 111. To illustrate the pro- cedure, a two-link planar robotic arm is optimized for strength and rigidity. The results are compared with those obtained by using the Kreisselmeier- Steinhauser function. 2. DESIGN PROBLEM In this section, the optimum design of a robotic arm is formulated as a nonlinear mathematical programming problem for strength and rigidity. The arm consists of N number of flexible links each of which are discretized by Ek number of beam finite elements. The objective is to minimize the weight of the arm. The contraints related to strength are the element stresses and the constraints for rigidity are the deviations of the selected points from the path of the rigid model. The design variables are the cross-sectional properties of the link elements. 255 2% S. Oral and S. Kemal Ider Mathematically, this is written as reduced stiffness matrices of & in terms of modal minimize the objective functionf = 5 2 pkP variables. To evaluate the submatrices in eqns (4, 5), 7” and /I” are expressed in the following form as: 1=I ,=I subject to constraints g,(x, t) 0 P, r = 1, 2, 3 j = 1, . , NC, Y:; = $, + $;,& (1) q=l,., n, s=l,., 12 (7a) where pkf and vk are the mass density and volume of the ith element of kth body, respectively, x is the fii;=8&+&,4!: p,r= 1,2,3 vector of NV number of design variables and N, is the total number of time-dependent constraints. In q= l,., m s= l,., 12, (7b) evaluating the displacements and stresses, the following recursive formulation based on Ref. lo is where 4” is the element shape function, n, is the employed to model the coupled rigid-elastic motion number of joint variables and m is the number of of the arm. modal variables. Note that in the equations, a Let the deformation of a link & be defined relative repeated subscript index in a term implies sum- to a link reference frame 5” which follows the global mation. Superscripts are generally part of the labeling motion of Bk in a manner consistent with the and do not imply summation unless otherwise boundary conditions. The number of elastic degrees- specified. The mass submatrices can be written as of-freedom of each link is reduced by modal reduction. Mz = f 2 mkjfjqj$, The generalized coordinates of the system are the k=,=, joint variables Bi and modal variables q, The velocity of a particle P, vki, can be written as + (7jqfi:z3 + 7i,j$sEi + j$,j$, RL.1 (84 M; = f 2 m&, k=,i=l where yk and /?“I are the corresponding influence coefficient matrices. Kane et al.s equations 12 are used to determine the equations of motion as Mjl=Q+F”+F, (3) + &a + !;,Y;X: + &L.xl (8) where y = dT, d is the vector of generalized speeds, where F is the vector of generalized applied forces, and M, Q and F are the generalized masses, Coriolis and centrifugal forces and elastic forces, respectively, as pki = UI s p”& d V and R:L$,. = s pkc#&$:.d V; v!- Vk shown below: F”=- O I Q!, = kf, ,$, mki/%4 K1 (6) + (/?&b;:, + u;&).f + &,b;:,.R:;,. (9b) where the superscripts r and f refer to rigid body and The equations of motion are integrated by using elastic degrees-of-freedom, respectively. K is a block a variable step, variable order predictor-corrector diagonal matrix whose diagonal submatrices are the _ algorithm to obtain the time history of the z,u= 1,2,3; s,v= l,., 12 are the time-invariant matrices, and mk is the mass of ith finite element of the kth body. By defining L = $?A& + & and bZ, = $!&,& + $&,&, the Coriolis and centrifugal forces can be computed as High-speed flexible robotic arms 251 generalized coordinates Bi and vi. Then nodal displacements with respect to the body reference frames are obtainfed by the modal transformation of vi. The element stresses are computed by the stress-displacement relations. The displacements of the points of interest in the global reference frame are found by using 0, and the nodal displacements in the body frames. The deviation of a point is defined as the difference between the global displacements of that point in the flexible and rigid models. It should be noted that, in the equations of motion, the only terms that are functions of design variables are the stiffness matrix, the element masses and the arrays Pk and Rk in the mass matrix and load vector. Hence in the analytical sensitivity analysis, these are the terms that should be differentiated with respect to the design variables. However, analytical evalu- ation of the sensitivities is a difficult task in this class of problems. A semi-analytical or overall finite difference approach is much better suited. 3. CONSTRAINT REDUCTION The dynamic response of the arm is calculated at N, number of discrete points in the time domain. Hence, the number of constraints to be satisfied becomes NC x N, and such a large number of constraints is not practical in an optimization process. An effective approach to keep the number of constraints as NC and to ensure satisfaction of constraints for all values of t is to define equivalent time-independent constraints by using Kreisselmeier- Steinhauser function 3 as g,(x) = - i In ? exp(-cg,) (10) .=I where gjn(x) = gj(x, t”) and c is a user-selected positive number which determines the relation between & and the most critical g, i.e. min(g,“). It can be shown that the Kreisselmeier-Steinhauser function defines a conservative envelope 4 such that gj is always more critical than min(g,n), and the larger the value of c, the closer & follows min(g,). This suggests using the most critical constraint as the equivalent constraint as Ej(X) = mingjn(x)l. (11) In this approach, the equivalent constraint gj defines a piecewise-smooth function with finite discontinuous gradients as it makes transitions from gjp to gjg. In this envelope, although the right- and left-hand deriva- tives are different at the transition points, they are of the same sign and the gradients are blended at the transition points by the numerical differentiation. In the limit as the time step approaches zero, the equivalent constraint becomes smooth. The nonlinear, constrained optimization problem defined above is solved by using the optimizer NLPQL l I which is based on sequential quadratic programming. This optimizer requires first-order information df/dx, and dgj/dxm, M = 1, . . . , NV, which are computed by overall finite differences in the present work. 4. NUMERICAL EXAMPLE A two-link planar robot is shown in Fig. 1. A single task is considered in which the end-effector E is required to deploy from an initial position (0, = 120”, 19 = - 150) to a final position (0, = 60”, e2 = - 30”) along a straight line. The prescribed motion of E is given as Ax =Ay =g T 2nt E E T t - x sm 7 The period of the deployment motion, T, is taken to be 0.5 s. Each link is of length 0.6 m and is modeled by two equal length tubular Euler beam finite elements. The outer diameters, &, k = 1, 2; i = 1,2 of the elements are taken as the design variables. The wall thickness of each element is set to be 0.1 Dni. The material properties are E = 72 GPa and p = 2700 kg rnm3. The problem size is reduced by using modal variables. The first two bending modes and the first axial mode with fixed-free boundary conditions are considered. The Fig. 1. A planar robotic manipulator. 24.0 22.0 t t 20.0 & 18.0 f 16.0 14.0 12.0 0 5 10 15 20 25 30 35 Number of iterations Fig. 2. Design histories. 258 S. Oral and S. Kemal Ider Table 1. Optimum solutions for the planar robotic manipulator KS-10 KS-30 KS-SO MCC Weight Dll 012 DZI 022 Number of (N) (mm) (mm) (mm) (mm) iterations 21.374 62.635 50.982 45.107 30.927 14 16.800 55.995 45.409 39.266 27.172 19 16.286 55.210 44.742 38.524 26.736 19 15.719 54.266 44.150 37.552 26.315 38 actuator of link-2 is located at joint-B has a mass of 2 kg and the combined mass of the end-effector and payload is 1 kg. The design problem is solved under the following constraints: -75MPaai75MPa i=l,.,n, 6 0.001 m, where the stress constraints are evaluated at n, number of points which are the top and bottom points at each node. 6 is the deviation (magnitude of the resultant of deviations in x and y directions) of the end-effector E from the rigid motion. The initial design is 50 mm for all design variables, Dki. In this example, the equivalent constraints are formed by employing the most critical constraints and the results are compared by using the Kreisselmeier-Steinhauser function. In the latter, different values of c have been tried. It has been observed that the lower values of c resulted in highly conservative designs, as expected. A value of c = 50 yielded a satisfactory design. It should be noted that the compiler limits may be exceeded for large values of c due to the exponential function if the lower bounds on design variables are set too small. On the other hand, the most critical constraint approach resulted in the lightest design satisfying the deviation constraint exactly. The minimum weights, optimum diameters and number of iterations are tabulated in Table 1. The design histories are shown in Fig. 2. The labels KS-c denote the results obtained by the Kreisselmeier-Steinhauser function, whereas MCC denotes the use of most critical constraint approach. It is seen that the stresses are far below the allowable 10.0 - KS10 - KS30 - KS50 -MCC 6.0 J 0.0 0.1 0.2 0.3 0.4 0.5 t w Fig. 3. The stresses at the middle of link-2 at the top in the optimum designs. 0.8 E 0.6 s P $ 0.4 0.2 Fig. 4. The end-effector deviation in the optimum designs. High-speed flexible robotic arms 259 values, hence the stress constraints are inactive. The stresses at the middle of link-2 at the top, where the maximum stresses occur, are plotted in Fig. 3. The end-effector deviation 6 for the optimum solution is shown in Fig. 4. 5. CONCLUSIONS In this study, a methodology for the optimum design of high-speed robotic manipulators subject to dynamic response constraints has been presented. The coupled rigid-elastic motion of the manipulator has been considered. The large number of time-de- pendent constraints has been reduced by forming equivalent time-independent constraints based on the most critical constraints whose time points may vary as the design variables change. It has been shown that the piecewise-smooth nature of this equivalent constraint does not cause a deficiency in the optimization process. Sequential quadratic program- ming is used in the solution of the design problem with sensitivities calculated by overall finite differ- ences. A high-speed planar robotic manipulator has been optimized for minimum we