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滚轴丝杆系统的运动学优化D. Mundo，机械系，大学的卡拉布里亚， 87036（政务司司长） ，意大利H.S. Yan，机械工程学系，国立成功大学，台南70101 ，中华民国摘要：本篇文章提出了一种方法来对传动机构进行运动学优化，过去常常通过非圆形的齿轮进行机械系统的控制来完成运动的输出。通过调查研究，本文将运用滚轴丝杆系统进行机械系统的控制来完成运动的输出。设计的目标函数是通过设计一对变半径的齿轮作为输入机构，从而使得螺杆的加速度峰值最小。为了去进行机构的优化问题，滚轴丝杆系统的运动学参数需要借助于无量纲的运动学方程来分析。遗传算法也被运用来进行目标函数的优化，并且在设计方法中添加了惩罚函数。最优机制的运动学分析发现了在恒定速度上带有恒定螺距螺杆的螺丝加速度有37 削减峰值。运动学仿真就是用来验证该方法的。关键词：滚珠丝杠传动;运动学优化;遗传算法;数控齿轮1. 引言在业界追求高生产力和高品质敦促着研究者去探讨有效的机制设计方法，以提高机器自动操作性能。提高输出运动特性的传统方法是假设输入速度恒定，并建议重新设计和制造不同的机制，更好的运动或动态表演。其中一个例子是Mills etal提出的柔性凸轮机构的优化设计。处理问题的不同办法就是积极控制机制的输入速度，设计了一个可变输入/输出驾驶系统。1956年，Rothbart 建议使用一种withworth快速回报机制，以提供一个凸轮可变输入速度，从而减少凸轮尺寸，压力角。后来，特萨和马修为分析变量投入高速凸轮从动机制导出运动方程。 变速装置的快速发展及伺服控制系统表明了，研究者应该设计伺服一体化机制，其特点是输入速度由计算机控制。 1994年，Chew and Plan用dcservo电机减少残余振动高速机电在机器的残余振动，而Yanetal发现了依靠凸轮的速度曲线的运动学特征。此外，他们提出了通过开发设计方法优化电脑控制和输入速度来主动控制凸轮机构的理论。1990年， kochev提出了在平面连接老化处积极平衡摇矩和扭矩波动，而最近Yaoetal研究了动态变速平面机制。作为一个平均数运动优化，尽管有广泛的文献关于可变输入速度函数，很少研究重点应用这一技术的滚珠丝杠传动。这种机制，基本形成了由滚珠丝杠连锁带动滑块曲柄系统，是用在几个工业应用。其中一个例子是用于纺织机械的螺杆传动机构。由于改善了凯恩斯-马泰行为，可变螺距螺丝常用于商业应用。 1993年，Yan and Liu提出了设计和制造可变螺距，导致了螺钉啮合圆柱的组成。他们还提出了三次多项式关系直线位移滑块和旋转螺丝。最近，Liu etal用一个伺服曲柄滑块机构积极控制输入速度，以降低峰值加速度的螺丝。本文介绍的工作目标是优化输出运动滚珠丝杠透射外，设计一个驱动机制，来基本形成了一个滑块曲柄驱动系统有一对非圆齿轮。然后提出联合机制，其中输入是不断旋转的速度驾驶数控齿轮，螺丝是被迫迁往最优运动。纯机械控制螺杆是基于运动学合成的可变螺距半径线，起始于最优输入/输出关系。自从一个灵活的控制不需要这种应用，一个便宜而有效的一双数控齿轮可取代各类计算机控制伺服。为了设计一个最佳的驱动机制，非维运动方程被导出了。然后确定设计约束作为刑罚功能插入的目标函数，而优化问题，是通过进化理论解决的。遗传算法被广泛使用于涉及全局优化问题。主要优点进化技术简单，在执行程序的数值及其很低的成本计算。此外，深层知识的数学特征搜索空间是不需要的。一旦数控齿轮的优化设计出来，虚拟样机联合机制和运动学仿真将会用来验证所提出的控制策略。2. 运动方程滚珠丝杠传动基本由两个合并机制形成：滚珠丝杠的联系通过曲柄滑块机构带动。曲柄旋转是机械系统的输入，而往复旋转螺杆是输出。图1机制是由五个部分组成：连杆1是基座，连杆2、 3、4，分别担负曲轴，连杆和滑块的驱动机制，连杆5是螺丝杠。因次运动方程可以通过考虑两个基本机制来推导，如图2所示图2（a） ，表达出了曲柄滑块机构的位置方程有以下几方面：上式中，r2、 r3分别为曲柄和连杆的长度， 2、 3分别为曲柄和连杆及r2、 r3杆与X轴之间的夹角。 联合（1）和（2） ，位移s的滑块，可确定为 （3）当曲柄旋转角度为角度时，所用的时间为机构运动的半个周期，滑块运动了2r2.的距离，则可以根据关系式定义出下面几个表达式：（4），（5），（6）联合（4）和（6），并把结果带入（3），得出，位移s的滑块，可确定为（7）这儿把 r3/2r2.定义为R3。分别对公式（7）对变量T进行微分，分别得到了滑块的速度、加速度、和加加速度的表达式：（8），（9），（10）参考 2（b） ，并假设螺丝杆斜度P为常数，输出旋转为（11）从而螺丝杆的旋转可以进一步的表达为：（12）因此，自旋转恒定螺距螺杆是沿着滑块的位移，螺钉和滑块量纲运动方程都是一样的。螺杆的速度、加速度、和加加速度的表达式（8） -（10）因次运动方程表明运动学恒螺距螺杆球变速器，根据最优运动规律的h（t）可迫使曲柄按照要求的角度来旋转。在以后有优化策略将被实施以设计一个最优控制功能。一旦非维运动特性的优化螺杆决定，但实际运动曲线可以由以下几个关系来确定（8）-（10）（13），（14），（15），（16）在这儿 , v, a ，j分别为滑块的角位移，速度，加速度，加加速度3. 优化控制方案最优控制螺杆运动要求目标函数和一套给出的设计规则。一旦优化问题被制定出来了，遗传算法将用来减低成本函数。刑罚方法确保最优解满足设计规则。3.1制定优化问题优化问题的主要目标是设计一个旋转速度功能曲柄以积极控制螺旋运动，并尽量在滑块前进冲程中减少其峰值加速度，使得在工作期惯性载荷问题可以减少。然而，无论是螺钉和曲柄运动特性应履行一套运动学要求和一般设计规则。螺杆速度和加速度变化曲线必须连续，其值必须满足要求。 因此，曲柄旋转速度x（t）必须至少有一个二阶可微函数是从输出（7） -（10）得到的。如果曲柄旋转的h（t）被选择作为一种控制功能，在第四或高阶多项式表达可以界定，然后在下列形式： (17)根据减少其峰值加速度为目标函数，设计变量为a0, . . ., aN已经确定出来了，下面根据要求来确定约束条件，并且约束条件必须满足公式（6）的要求：(18),(19)根据公式(17)(19)，约束条件为：(20),(21)为了完成制定的优化问题，成本函数必须界定。主要目的是为了减少螺丝的加速度峰值。然而，优化控制功能，必须履行下列设计规则： 1.一个典型的曲柄旋转不断改变方向。因此，时间导控功能不能改变迹象。不丧失普遍性，在这项工作中曲柄速度将保持正的。2.第二次导控功能必须温和，因为非圆齿轮将用来提供有变速功能的曲柄和将决定不规则线间距的角度的突然变化值。在设计的要求的基础上，价值函数能够被定义为（22）当有一个或更多的变化，周期p用来惩罚控制功能时，根据不同的优化策略，权值w1和w2可以调整。处罚方法保证了设计的实现，因为任何一套行不通的设计参数比一个容许的解决方案会有更大的价值。明显的，如果下式成立，那么公式（22）中的惩罚函数应该被置为0.3.2 优化方法 为了解决在上一节基础上制定的优化问题，一种改进的方法被使用了。一种改进的遗传算法，如图3所示。第一步设计开始的数目，形成Np对个体。每个组成有一套设计容许的变量值。因此，一般每个个体是解决优化问题的一个可能，可以作为真实数字形式的一个载体。其中n是一些独立设计变量进化优化方案使得那些满足要求的优化结果被保留了下来。这些要求的优化结果经历了一套遗传操作，以使得在以后的运算中能够继续保留下来。这个过程称为自然选择。再生产的第一步是复制过的Np对个体的选择，其遗传信息将合并产生新的Np对个体。因此，个体规模是保持恒定的。可以基于不同的概率分布，包括均匀分布来挑选个别复制。标准化几何排序选择方法用于这项工作的算法中。每个个体按照他们的特点不同，不被选到的可能性 ，按下列表达（24）Pb是一个恒量，是与选择最佳个体的概率成正比的， r是个体的一个排列。排列1是最佳个体， Np是最差的一个选择。因此，进行筛选的过程中，成本函数中的每个个体必须评估以保证最后选择的是最好的。一旦两个体X1和X2被选定复制，遗传操作（交叉）会产生一个新的个体X1，其基因是来自原有X1和X2的一些特点。在这项工作中的新的个体所创造的手段启发式交叉操作如下：（25）而rn是一个实数,随机选取范围 0,1 根据精密计划，新个体中有比在他之前的两个组合更加优越才会进入下一选择。否则新个体将会被拒绝，X1和X2之间的最佳染色体会被保留。 复制的最后一步是突变，变更新个体的部分遗传信息。突变是必要的，以防止算法收敛趋近当地的最低条件，是、而且突变需带有概率 PM 2 0,1一起进行 。在这项工作中非均匀突变也是要利用的。个体基因的突变是按下列计划随意改变的：（26）ri是范围 0,1 内的随机数 ; ui 和 li是基因边界上和下界的;f（g）是一个函数，定义如下：（27）其中rn2范围 0,1 内的一个随机数，b是形状变化参数，G是循环次数； Gmax是最高迭代次数。 序列健身评价，选择个人复制，交叉和变异是迭代，根据该图3所示，直到人数最多的几代人出现或实现低成本。3.3 优化控制函数在开始进行程序的迭代之前，遗传参数必须初始化。开始个体数目的选择应该建立数字化设计变量的基础之上。 在应用中存在着被选为控制功能的第七次多项式函数。因此，根据均衡方程式 （17），（20）和（21） ， 6个设计变量必须运用进化理论。开始的个体数目是 60个。设置Pb = 0.5, PM =0.1,Gmax=100和b=0.85 ，最佳个体数目被认定为如下（28）优化控制功能，即因次曲柄旋转由平衡方程（17）-（21）和最佳个人基因所决定，就是：如图4所示，从优化的过程中得出了曲柄的位移，速度，加速度和加加速度曲线，由 OPT(T)和其对时间的导数方程式 （7）-（10）得出 ，螺丝杆的最佳曲线量纲角位移，速度，加速度和加加速度的曲线。在如图5中，进行了辊轴丝杠为变速度的输入进行了优化，并将这些曲线与辊轴丝杠为恒定速度的输入相比。当以一个变量投入高速传输时，机构的动力学特点和文献Liu 12得出的结果基本上是一致的。在表1中，总结出来了在不同的结构参数组合下，得到了螺杆的位移、速度、加速度、及加加速度曲线，在表1中并对不同的结构参数组合曲线的峰值进行了比较。目标函数优化的主要是问题，如公式3所示 ，降低峰值加速度的螺丝。从表1中可以看出，辊轴丝杠为变速度的输入与辊轴丝杠为恒定速度的输入相比，螺丝加速度的峰值减小了36.9%，而当使用伺服控制机械系统时，螺丝加速度的峰值减小了19.4%。而使用上面的两种方法时，螺杆的速度、加加速度与原来的比较都有所减小，因此上面提出的机构的优化方法得到了证明。为了去评价本文所提出的优化算法的性能，优化目标函数，在图6中，找出了快速收敛至近最优解，最快的收敛算法和最好的接近目标函数的算法被找到了，并且在经过了仅仅68步迭代运算以后，函数值与目标函数值达到了98.9%。最终的结果被得到了在100步迭代的时候。因此，该算法能做出6000份合适的不同解决办法的评价，因为个体数是60。计算算法的时间，在AMD Athlon XP 3000上执行的，是54.32s。变机构的效果也经过了上面优化算法的几次检测，尽管在优化过程中，开始的各个值选择不同，但是，最后的解决方法基本上都是类似。因此，开始提出的目标函数能够得到很好摆脱当地的最低条件。 fig 64.驱动机制的设计为了使得曲柄旋转按照上面最优化的方案来运动，如图4所示，一对非圆齿轮可以用来代替更昂贵的电脑控制伺服系统。在此应用中，事实上，运用一对非圆齿轮这种具体的机械控制螺杆运动学是有效的，因为灵活的控制系统，在这里是不需要的。 设计一对非圆齿轮是基于输入/输出关系的要求。在这个应用中，持续的旋转速度假定为传动齿轮的转速，而从动齿轮，必须按照优化曲柄运动规律所决定的来运动，如公式（29）。此外，虽然机械控制螺旋运动在前一段行程的冲击中是必要的，但是，利用非圆齿轮在后一段行程的冲击设计中也是需要的。因为不包括具体的要求，履行连续性传输的唯一条件，曲柄后段行程的设计中，可以用多项式表达，如下：（30）a0到a7必须依靠下面的几个表达式定义：（31），（32），（33），（34）一旦曲柄的两个行程被定义了，变量齿轮的函数表达式如下：（35）Xout是从动齿轮角速度，Xin = 1是传动齿轮输入速度的量纲恒速。图7显示可变齿轮比规定了具体控制任务以及与此相关的应用程序。注意到，在结果（31）中，齿轮比的传动比为一。 fig 7 fig 8综合的螺杆线斜度能够按照以下的公式定义： （36），（37）其中（T）= T是传动齿轮的因次旋转，R1和R2是从动齿轮的可变半径，D是轴心之间的距离，其轴线如图8所示。结合公式（36）及（37），在应用中，D设置为100毫米，可变螺距半径在图9（a）进行了合成的。设计过程的最后一步，是齿廓。在本文的数学模型中，齿廓是基于以下微分方程19 ： （38）d是传动齿轮旋转的角速度，dy是位移值，即组成相应的接触点沿位移线的位移，是一个压力角，沿各剖面保持恒定。为了产生第一个配对的齿轮副，当前接触点的坐标值必须由公式（38）计算得出 ，从任意一个齿形的交点开始。基于两个matrices旋转矩阵，角度和h（）包含在内进行坐标变换，然后进行操作，获得两个不同点，从当前的接触点，进行对共轭分布。当下列情形之一发生时，这个综合的优化过程将停止下来，：径向轮齿的尺寸超过既定的增值;距离目前联络点和齿轮中心固定。最后条件，实际上意味着削弱。这个数值程序，然后将这个数值程序运用到每一个轮齿上，下面已知直径螺距和齿轮数的基础上，对第一个啮合点进行修改。压力角也进行了一定的修改，使该夹角压力线与螺距曲线的正常线保持恒定。显然，对不同齿轮设计时，两啮合齿要假设同一压力角。在如图9（a）中所示，假设齿数nZ = 35，平均值压力角等于20,在图9（b）中产生了齿轮.5.虚拟样机及运动仿真虚拟运动学分析是对模型进行运动仿真，以验证提出的优化方法。列图10中所示 ，一个结合机制的三维模型，是借助计算机辅助软件，PTC- ProEngineer 造出的。虚拟滚珠丝杠传动列的规格在表2中列出了。结合机制的运动学是通过相同软件的机制模块模拟的。驱动齿轮轴提供一个恒速驱动，而从动齿轮根据最优运动规律转动，由公式（29）及（30）中确定。滚珠丝杠传动的曲柄是根据运动学曲线图4被迫运动。用两个旋转接头连接杆和曲柄以及杆和滑块，而后者则是用一个圆柱副连接螺杆。第三旋转副把螺杆与地面的框架连接在一起。为了再现滚珠丝杠传动的运动学，使关系得以确立，根据公式（11）中，螺杆旋转和滑块线性位移（螺杆联合），两者的关系需要设立。在对虚拟样机的运动学仿真时，测量了螺丝杆的运动学特征。结果如图11 ，如图所示，对没有进行优化时的螺杆恒定输入和优化后的机构进行了运动学特性的比较。螺杆加速度曲线不断输入速度配置显示峰值4693.4 rad/s2 ，而最高值3002.3 rad/s2已计算出作为最优机制。因此，运动学仿真证实了最优控制的结果，从而验证方法的有效性。6.结论本篇文章提出了一种方法来对滚轴丝杆传动机构进行运动学优化，优化控制的方法已经被应用了，即通过设计一对变半径的齿轮来作为输入机构来进行了机械系统的控制来完成运动的输出。首先，通过对机构的运动学分析，得到了一些无量纲的运动学方程，通过机构的优化来使得螺杆的加速度峰值减小，当机构的加速度峰值减小了，则机构在工作行程中的机构运动的惯性力也就减小了，工作也就更稳定了。运动学分析，要求对双方，即输入和输出机制，进行分析和考虑，同时还要运用到界定的控制功能和实施优化程序中去。进化理论一直作为一种全局优化技术用来运行遗传算法，同时用处罚法来进行程序的优化，这样就使得优化法变为了一个可行的解决办法。提出控制方法的最后一步，是在最优解的基础上，设计出一对非圆齿轮。通过分析结合机制的运动学，提出的最优控制策略表明其能有效地改善输出运动特性。在不断的投入高速螺旋传输中，螺丝加速度峰值减小了36.9%，最终的结果也通过机构的运动学仿真得到了证实。由于通过简单的组合机构而达到了很好的效果，作者认为本文提出的这种最优控制方法，对于设计和制造工业涉及机械螺杆传动机构，特别是重载应用，是一个重大的贡献。Kinematic optimization of ball-screw transmission mechanismsD. Mundoa,*, H.S. YanbaDepartment of Mechanical Engineering, University of Calabria, 87036 Arcavacata di Rende (CS), ItalybDepartment of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROCReceived 20 June 2005; accepted 13 February 2006Available online 4 April 2006AbstractThe paper proposes a method for the kinematic optimization of transmission mechanisms, where non-circular (NC)gears are used to perform a mechanical control on the output motion. The investigation presented here deals with themotion control of a ball-screw transmission mechanism. The objective is lowering the peak acceleration value of the screw,by designing a pair of variable radius gears as a driving mechanism. The kinematic characteristics of the ball-screwmechanism are analyzed by means of non-dimensional motion equations in order to formulate the optimization problem.A genetic algorithm (GA) is then implemented to optimize the objective function, and a penalty method is used to fulfil thedesign rules.The kinematic analysis of the optimal mechanism revealed a 37% reduction of the peak acceleration of the screw incomparison with a constant pitch screw, operated at a constant speed. A kinematic simulation is used to validate themethod.? 2006 Elsevier Ltd. All rights reserved.Keywords: Ball-Screw transmission; Kinematic optimization; Genetic algorithm; NC gears1. IntroductionThe pursuit of high productivity and high quality in industry urges researchers to investigate on effectivemechanism design methods in order to improve the performances of automatic machines.The traditional methods of improving the output-motion characteristics assume the input-speed to beconstant and propose to redesign and to manufacture a different mechanism, with better kinematic or dynamicperformances. An example is the optimal design of flexible cam mechanisms proposed by Mills et al. 1.A different approach to the problem is the active control of mechanism input-speed, by designing a variableinput/output driving system. In 1956, Rothbart 2 proposed the use of a Withworth quick-return mechanismto provide a cam with a variable input-speed, thus reducing the cam dimensions and hence the pressure angle.Later, Tesar and Matthew 3 derived motion equations for the analysis of variable input-speed cam-followermechanisms.0094-114X/$ - see front matter ? 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2006.02.002*Corresponding author. Tel.: +39 0984 494159; fax: +39 0984 494673.E-mail address: d.mundounical.it (D. Mundo).Mechanism and Machine Theory 42 (2007) 3447/locate/mechmtMechanismandMachine TheoryThe rapid development of servomotors and their control systems suggested researchers to design servo-inte-grated mechanisms, characterized by a computer controlled input-speed. In 1994, Chew and Plan 4 used a dcservomotor to minimize the residual vibrations in high-speed electromechanical bonding machines, while Yanet al. 5,6 demonstrated that the kinematic characteristics of followers are dependent on cams speed curve.Furthermore, they proposed the theory of Active Control of cam mechanisms 7, by developing a method todesign the optimal computer-controlled input-speed function.In 1990, Kochev 8 proposed to actively balance shaking moments and torque fluctuations in planar link-ages, while recently Yao et al. 9 studied the dynamics of variable-speed planar mechanisms.In spite of a wide literature about variable input-speed functions as a mean of motion optimization, fewresearches focus on the application of this technique to ball-screw transmissions. Such mechanisms, basicallyformed by a ball-screw linkage, driven by a slidercrank system, are used in several industrial applications. Anexample is the screw transmission mechanism used in textile machines 10. Because of their improved kine-matic behaviour, variable pitch screws are commonly used in commercial applications. In 1993, Yan andLiu 11 proposed a method to design and to manufacture variable pitch lead screws with cylindrical meshingelements. They further suggested a cubic polynomial relationship between the linear displacement of the sliderand the rotation of the screw. Recently, Liu et al. 12 used a servomotor to actively control the input-speed ofthe slidercrank mechanism, in order to reduce the peak acceleration of the screw.The objective of the work presented in this paper is to optimize the output motion of ball-screw transmis-sions, by designing a driving mechanism, basically formed by a slidercrank system driven by a pair of non-circular gears. A combined mechanism is then proposed, where the input is the constant rotating speed of thedriving NC gear, and the screw is forced to move according to an optimal law of motion. The pure mechanicalcontrol of the screw is based on the kinematic synthesis of variable-radius pitch lines, starting from the optimalinput/output relationship 1315. Since a flexible control strategy is not required in this application, com-puter-controlled servomotors can be replaced by a cheaper and effective pair of NC gears.In order to design an optimal driving mechanism, non-dimensional motion equations are derived. Theobjective function, in which the design constraints are inserted as penalty functions, is then defined, whilethe optimization problem is solved by using evolutionary theory 16. Genetic algorithms are widely used inproblems involving global optimization. The main advantages of evolutionary techniques are their simplicity inimplementing the numerical procedure and their low computational cost 17. Furthermore, a deep knowledgeof the mathematical characteristics of searching space is not required.Once the optimal design of NC gears is performed, a virtual prototype of the combined mechanism and akinematic simulation are used to validate the proposed control strategy.2. Motion equationsA ball-screw transmission is basically formed by two combined mechanisms: a ball-screw linkage, driven bya crankslider mechanism. Crank rotation is the input of this mechanical system, while the reciprocatingrotation of the screw is the output. In Fig. 1 the mechanism is schematically represented as consisting of fivemembers: link 1 is the frame, links 2, 3 and 4 are, respectively, the crank, the connecting rod and the slider ofthe driving mechanism, link 5 is the screw. Dimensionless motion equations can be derived by considering theFig. 1. Schematic representation of a ball-screw transmission mechanism.D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344735two basic mechanisms separately, as shown in Fig. 2. By referring to Fig. 2(a), the position equations of theslidercrank mechanism are the following:r3cosh3 r2cosh2 r4;1r3sinh3 r2sinh2;2being r2and r3the lengths of the crank and of the connecting rod, h2and h3the angles the links form with thenegative X-axis, r4the position of the slider.By combining Eqs. (1) and (2), the displacement s of the slider can be determined ass r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir3r2? ?2? sin2h2s? cosh2?r3r2? 1?2435.3While the crank rotates an angle p in a time s equal to half a period of the mechanism motion, the slidercompletes a stroke of 2r2. Dimensionless motion equations can be then derived by defining the following non-dimensional parameters:T ts;4S s2r2;5H h2p;6where t 2 0,s, s 2 0,2r2, h 2 0,p, while T, S and H vary between 0 and 1.By substituting Eqs. (4)(6) in Eq. (3), the non-dimensional slider displacement is obtained:S 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2R32? sin2pHq? cospH ? 2R3? 1? GH;7where the non-dimensional length R3is defined as r3/2r2.Fig. 2. Schematic representation of the crankslider mechanism (a) and of the ball-screw linkage (b).36D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447By differentiating Eq. (7) with respect to T, the dimensionless slider velocity (V) and then acceleration (A)and jerk (J) are obtained:V dSdT G0H ?_H;8A d2SdT2 G00H ?_H2 G0H ?_H ?H;9J d3SdT3 G000H ?_H3 3G00H ?_H ?H G0HvH.10By referring to Fig. 2(b) and assuming the pitch p of the screw to be constant, the output rotation / is givenby/ 2pps;11from where the dimensionless screw rotation r can be derived asU /max S.12Therefore, since the rotation of a constant-pitch screw is linearly dependent on the slider displacement, thescrew and the slider dimensionless motion equations are the same. The screw non-dimensional velocity, accel-eration and jerk can be then computed by means of Eqs. (8)(10).Dimensionless motion equations show that the kinematics of constant-pitch screw-ball transmissions canbe improved by forcing the crank to rotate according to an optimal law of motion H(T). In the following sec-tions an optimization strategy will be implemented in order to design an optimal control function.Once non-dimensional motion characteristics of the optimal screw are determined, the actual kinematiccurves can be determined by the following relationships, deriving from Eqs. (8)(10):/ 2pp? 2r2? S;13v d/dt2pp? 2r2?dSdt2pp? 2r2?dSdT?dTdt2pp?2r2s? V ;14a dvdt2pp?2r2s?dVdt2pp?2r2s2? A;15j dadt2pp?2r2s2?dAdt2pp?2r2s3? J;16where /, v, a and j are the actual angular displacement, velocity, acceleration and jerk of the screw.3. Optimal control strategyThe optimal control of the screw motion requires that the objective function and a set of design rules aredefined. Once the optimization problem is formulated, a genetic algorithm will be used to minimize the costfunction. The penalty method is employed to ensure the optimal solution fulfils the design rules.3.1. Formulation of the optimization problemMain objective of the optimization problem is to design a rotating-speed function of the crank in order toactively control the screw kinematics and to minimize its peak acceleration during the forward stroke of theslider, so that inertial loading problems can be reduced during the working period. Moreover, both the screwand the crank motion characteristics should fulfil a set of kinematic requirements and general design rules. Thescrew velocity and the acceleration curves must be continuous, while finite values of the jerk are required.Therefore, it is known from Eqs. (7)(10) that the crank rotating speed X(T) must be at least a second orderD. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344737differentiable function. If the crank rotation H(T) is selected as a control function, a fourth- or higher-orderpolynomial expression can be then defined in the following form:HT a0XNi1aiTi;17where the design variables (a0, ., aN) must be determined so that the screw rotates according to the optimallaw of motion and the following boundary conditions, deriving from Eq. (6), are satisfied:H0 0;18H1 1.19From Eqs. (17)(19), the following conditions are derived:a0 0;201 ?XNi1ai 0.21To complete the formulation of the optimization problem, the cost function must be defined. The mainobjective is to reduce the peak acceleration of the screw. However, the optimal control function must fulfilthe following design rules:1. A typical crank continuously rotates without changing direction. Therefore, the time-derivative of the con-trol function cannot change in sign. Without loss of generality, in this work the crank speed will be keptpositive.2. The second time-derivative of the control function must be moderate, since non-circular gears will be usedto provide the crank with the variable-speed function and sudden changes in_HT would determine irreg-ular pitch lines.On the basis of the design rules, the cost function can be defined asCa0;.;aN w1maxabsA w2maxabsH P;22where the weighing factors w1and w2can be adjusted according to different optimal strategies, while the termP is used to penalize the control function when one or more changes in the sign of_H occur. The penaltymethod assures that the design constraints are fulfilled, since any infeasible set of design parameters will havea greater value of the cost function than an admissible solution. Obviously, if_HT 0 T 2 0,1, the penaltyterm in Eq. (22) is set to zero.3.2. Optimization methodIn order to solve the optimization problem as formulated in the previous section, the strategy of evolution-ary methods is used. A modified genetic algorithm, schematically represented in Fig. 3, is employed in thispaper 18. The first step is the generation of the starting population, formed by NPindividuals (chromo-somes). Each individual consists of a set of design-variable admissible values (genes). Therefore, the genericindividual is a possible solution of the optimization problem and can be represented by a vector of real num-bers in the formXi x1x2?xn?;i 1;.;NP;23where n is the number of independent design variables.The evolutionary optimization strategy is based on the survival of the fittest individuals. These individualsundergo a set of genetic operations (reproduction) in order to promote the population evolution. This processis known as natural selection. The first step of reproduction is the selection of NPcouples of individuals (par-ents), whose genetic information will be combined to generate one individual (child) for the next population.38D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447Therefore the population size is kept constant. Selection of individual for reproduction can be based on dif-ferent probability distributions, including uniform distribution. In the algorithm used in this work, the methodof normalized geometric ranking selection is used. The probability for an individual to be selected is based onits performance (fitness), according to the following expression:PrPb1 ? 1 ? PbNP1 ? Pbr?1;24where Pbis a constant and is proportional to the probability of selecting the best individual, and r is the rankof the individual. The rank is 1 for of the best individual, Npfor the individual with the lowest fitness value.Therefore, before selecting individuals for reproduction, the cost function of each chromosome must beevaluated in order to establish a fitness-based order inside the population.Once two individuals X1and X2are selected for reproduction, a genetic operation (crossover) is necessaryto generate a new individual X?1, whose genes are derived from parents chromosomes. In this work the newindividual is created by means of a heuristic crossover operation, as follows:X?1 X1 rnX1? X2;25where rn is a real number randomly selected in the range 0,1.According to an elitist strategy, the new individual will enter the next population only if it has a greaterfitness than its parents, otherwise X?1will be rejected and the best chromosome between X1and X2is retained.Fig. 3. Scheme of the genetic algorithm.D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344739The last step of reproduction is mutation, an operator that changes a piece of genetic information of thenew individual. Mutation is necessary in order to prevent the algorithm from converging towards a local min-imum condition, and it is carried out with a probability PM2 0,1. In this work non-uniform mutation isemployed. A gene of the mutating individual is randomly changed according to the following scheme:xi? xi ui? xifGif ri 0:5;xi? xi? lifGif ri6 0:5;?26where riis a random number in the range 0,1; uiand liare the upper- and lower-bound of the ith gene; f(G) isa function defined as follows:fG rn21 ?GGmax?b;27being rn2a random number in the range 0,1, b the shape parameter of mutation, G the current generation,Gmaxthe maximum number of iterations.The sequence of fitness evaluation, selection of individuals for reproduction, crossover and mutationis iterated, according to the scheme of Fig. 3, until the maximum number of generations is reached or acost-function low value is achieved.3.3. Optimal control functionBefore starting the iterative procedure as described in the previous section, GA parameters must be initi-alized. The size of starting population should be established on the basis of the number of design variables.In the application presented here a seventh order polynomial function is chosen as a control function. There-fore, according to Eqs. (17), (20) and (21), six design variables must be determined by means of evolutionarytheory. A starting population of Np= 60 individuals is then generated. By setting Pb= 0.5, PM= 0.1,Gmax= 100 and b = 0.85, the best individual of the final population was found to be the following:XBEST 1:1685?0:25711:2e?52:3984?5:90515:0918?.28The optimal control function, that is the dimensionless crank rotation as defined by Eqs. (17)(21) and bythe genes of the best individual, is the following:HOPTT 1:168 ? T ? 0:257 ? T2 2:398 ? T4? 5:905 ? T5 5:091 ? T6? 1:495 ? T7.29Fig. 4. Optimal kinematic curves of the crank.40D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447In Fig. 4 the curves of dimensionless displacement, velocity, acceleration and jerk of the crank are shown,as deriving from the optimization procedure.By substituting HOPT(T) and its time-derivatives in Eqs. (7)(10), the optimal curves of dimensionless angu-lar displacement, velocity, acceleration and jerk of the screw are obtained as a function of dimensionless timeT. In Fig. 5 these curves are compared with those obtained for a constant input-speed ball-screw transmission.The same figure shows the kinematic behaviour of a variable input-speed transmission, where the crank movesaccording to the rotating-speed function proposed by Liu 12.A numerical comparison between different configurations is summarized in Table 1, where the peak valuesof the dimensionless velocity, acceleration and jerk of the screw are listed.The main objective of the optimization problem, as formulated in Section 3, was lowering the peak accel-eration of the screw. Table 1 shows that the optimized kinematics is characterized by a 36.9% reduction of thisvalue over the constant input-speed screw-transmission, while a 19.4% reduction over the servo-controlledmechanism proposed in 12 is achieved. Moreover, both peak velocity and jerk of the optimized screw arelower than in the last configuration. The effectiveness of the proposed optimal control strategy is then proved.In order to evaluate the performances of the proposed optimization algorithm, the evolution of the goal func-tion, i.e., the cost function of the best individual at each iteration, is plotted in Fig. 6. The fast convergence ofthe algorithm to a near optimal solution is shown, since the value of the cost function is 98.9% of the final bestsolution after only 68 iterations. The final solution is found after 100 iterations. Therefore, the algorithm per-forms 6000 evaluations of the fitness of different solutions, since the population consists of 60 individuals. Thecomputational time of the proposed algorithm, executed on an AMD Athlon Xp 3000, is 54.32 s.The effectiveness of the mutation mechanism has also been tested by executing the algorithm several times.Although the individuals of the starting population are always different, being randomly chosen within aFig. 5. Kinematic characteristics of three different ball-screw transmission mechanisms.Table 1Kinematic characteristics of three configurations of a ball-screw transmissionPeak valuesOptimized kinematicsConstant input-speedVariable input-speed as proposed in 12Velocity1.741.681.80Acceleration?4.29?6.8?5.32Jerk?29.98?23.5?34.32D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344741defined range of values, the final best solution is always very similar. Therefore, the proposed algorithm is ableto escape from local minimum conditions.4. Design of the driving mechanismIn order to force the crank to rotate according to the optimal law of motion represented in Fig. 4, a pair ofnon-circular gears can be used instead of more expensive computer-controlled servomotors. In the applicationpresented here, in fact, a specific mechanical control of the screw kinematics is effective, since flexibility of thecontrol system is not required.The design of a pair of non-circular gears is based on the requested input/output relationship. In this appli-cation a constant rotating-speed is supposed for the driving gear, while the driven gear must rotate accordingto the optimal crank law of motion as determined by Eq. (29). Furthermore, although mechanical control ofthe screw motion is necessary only during the forward stroke, the use of non-circular gears requires that thebackward stroke is also designed. Since specific requirements are not involved, being continuity of transmis-sion the only condition to fulfil, the crank backward motion can be designed by means of a polynomial expres-sion, as follows:HT a0X7i1aiTi;1 6 T 6 2;30where coefficients a0, ., a7must be determined by means of the following boundary conditions:H2 2;H1 H1?;31_H2 _H0;_H1 _H1?;32H2 H0;H1 H1?;33vH2 vH0;vH1 vH1?.34Once both strokes of the crank are defined, the variable gear-ratio function can be expressed assT XoutXin_HT;0 6 T 6 2;35where Xoutis the driven-gear angular velocity, and Xin= 1 is the dimensionless constant-speed of the drivinggear. Fig. 7 shows the variable gear-ratio law required by the specific control task involved in this application.Note that, as a consequence of Eq. (31), the average value of the gear ratio is one.Fig. 6. Evolution of the goal function.42D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447The synthesis of the pitch lines can be now performed using the following equations:sW R1WR2HW;36R1W R2WH D;37where W(T) = T is the dimensionless rotation of the driving gear, R1and R2the variable radii of the drivingand driven gear, respectively, D the distance between their axes, as shown in Fig. 8.By combining Eqs. (36) and (37) and setting D = 100 mm in this application, the variable-radius pitch linesof Fig. 9(a) are synthesized.The last step of the design process is the generation of tooth profiles. In this paper the mathematical modelof tooth profiles is based on the following differential equation 19:dydW cos2a R1W ?dR1dWtana?;38where dW is the elemental rotation of the driving gear, dy is the Y-component of the corresponding contact-point displacement along the line of action, a is the pressure angle, kept constant along each profile.In order to generate the first teeth pair, coordinates of the current contact point are computed by numer-ically integrating Eq. (38), starting from an arbitrary intersection point of the tooth profile with the pitch line.Coordinate transformations, based on two matrices of rotation where angles W and H(W) are involved, areFig. 7. Gear ratio function as required for the control task.Fig. 8. Polar representation of the pitch lines.D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 344743then operated in order to derive two different points on the conjugate profiles from the current contact point.The integration process stops when one of the following conditions occurs: the radial dimension of a toothexceeds the established addendum value; the distance between the current contact point and the gear centreis stationary. The last condition, in fact, entails undercutting.This numerical procedure is then repeated for each teeth pair, by modifying the first integration point onthe basis of diametral-pitch and teeth-number values. The pressure angle is also modified, so that the anglebetween the pressure line and the normal-line to the pitch curves is kept constant. Obviously, two meshingteeth on different gears are designed by assuming the same pressure angle.By applying the method described above to the pitch lines of Fig. 9(a) and assuming a number of teethnZ= 35 and an average value of the pressure angle equal to 20?, the gears represented in Fig. 9(b) aregenerated.5. Virtual prototyping and kinematic simulationA virtual kinematic analysis is performed in order to validate the proposed optimization method. A three-dimensional model of the combined mechanism, shown in Fig. 10, is created by means of a CAD software,PTC-ProEngineer?. The specifications of the virtual ball-screw transmission mechanism are listed in Table 2.The kinematics of the combined mechanism is then simulated by means of Mechanism Module of the samesoftware. The driving gear axis is provided with a constant-speed driver, while the driven gear rotates accord-ing to the optimal law of motion, as determined by Eqs. (29) and (30). The crank of the ball-screw transmis-sion is thus forced to move according to the kinematic curves shown in Fig. 4. Two revolute joints connect therod to the crank and to the slider, while the latter is connected to the screw by means of a cylindrical joint. Athird revolute pair is introduced to connect the screw to the frame.In order to reproduce the kinematics of a ball-screw transmission, a relationship is established, according toEq. (11), between the screw rotation and the slider linear displacement (screw joint).Fig. 9. Non-circular gears proposed for the optimal control of ball-screw transmissions.44D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447The kinematic characteristics of the screw are virtually measured while the simulation is performed. Theresults are shown in Fig. 11, where the kinematic behaviour of both the optimal and the constant-inputball-screw transmission are represented.The screw acceleration curve of the constant input-speed configuration shows a peak value of ?4693.4 rad/s2, while a maximum value of 3002.3 rad/s2has been computed for the optimal mechanism. Therefore, thekinematic simulation confirms the effectiveness of the optimal control strategy, thus validating the proposedmethod.Fig. 10. Three-dimensional model of a ball-screw transmission driven by non-circular gears.Table 2Specifications of the ball-screw transmission as defined for simulationConstant input-speed200 rpmLength of the crank (r2)100 mmLength of the connecting rod (r3)280 mmScrew thread2Pitch of the ball-screw40 mmTotal length of the ball-screw300 mmFig. 11. Screw motion characteristics as resulting from the kinematic simulation.D. Mundo, H.S. Yan / Mechanism and Machine Theory 42 (2007) 3447456. ConclusionIn this paper a method for the kinematic optimization of ball-screw transmission mechanisms has beenproposed. An optimal control strategy was implemented, by designing a combined mechanism: a drivingmechanism, based on non-circular gears, has been employed in order to perform a mechanical control onthe output motion.At first, non-dimensional equations of motion have been derived by means of a kinematic analysis of themechanism. A global optimization problem has been then formulated, whose main objective was lowering thepeak-value of the screw acceleration, in order to reduce inertial loading problems during the working stroke.The kinematic requirements of both driving and output mechanisms have been analyzed and taken intoaccount while defining the control function and implementing the optimization procedure. Evolutionary the-ory has been used to implement a genetic algorithm as a global optimization technique, and a penalty methodhas been employed to promote the evolution of the population towards a feasible solution.As a last step of the proposed control methodology, a pair of non-circular gears were designed on the basisof the optimal solution.By analyzing the kinematics of the combined mechanism, the proposed optimal control strategy was provedto be ef