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齐 齐 哈 尔 大 学毕业设计(论文) 、 题 目 电视遥控器注射模具设计 院（系） 机械工程学院 专业班级 机械032班 学生姓名 陈 阜 东 指导教师 陈 集 成 绩 2007 年 06 月 17 日 () 032 2007 06 17 用于塑料注射模具设计和生产的自动基准尺寸摘要：基准尺寸（或者坐标尺寸）技术在表面有大量必须有规定的孔特征的注射模具画图方面应用非常广泛。尽管商业CAD/CAM系统提供了半自动工具来协助设计者进行尺寸确定，但它仍然是个非常复杂的过程。作为使用者，不得不进行规定每个尺寸标记的位置。这篇文章讲的是一种最优的尺寸标记的全自动方法。这种方法采用的是动态工程技术，在尊重用户所选择的标准前提下，使尺寸确定最优化。这种方法已经作为一种工具插到了商业CAD/CAM系统中，并且给出了一些实例来说明这个技术的重要特征。关键字：自动尺寸 基准尺寸 动态工程 最优尺寸 坐标尺寸1介绍 CAD/CAM系统现在在塑料模具注射制造业应用非常广泛，许多公司可用一种实体模拟系统来设计注射模具。他们用CAD系统来设计，不仅仅是核心和孔插入到模具（他们是形成模具最重要部分），同时也是在模具装配中其他所有部分。伴随着互联网技术的进步和最近CAD网络通信的发展，注射模具的设计信息可以在产品工程师（设计塑料部件）和刀具工程师（设计注射模具），甚至他们位于世界上不同区域。在设计信息通过电子方式在产品设计和刀具设计中可以有效的交流，制造信息通信在电子和传统技术共同作用下在车间完成。数控机床刀具轨迹或者检查指令可以直接从CAD/CAM和互联网下载给数控控制器进行加工或者进行检查工作。然而，对于一个专门机器来说建立一个说明书或许在工程图纸中规定。另外，不是所有的加工任务都是在数控机床上完成。出于成本预算的考虑，一些传统机床，例如钻床和磨床等，可以方便用传统刀具完成。通常工程图纸也在车间里的工程信息通信中扮演重要角色。正交投影工程图纸可以从CAD模型中自动产生。零件尺寸自动工具同样由商业CAD系统提供。然而，就像有chen指出的，那些尺寸工具不能够根据图纸标准和工厂通常采用的工程方法来生成尺寸。在注射模具的特殊要求中，孔特征的基准尺寸（或坐标尺寸）应用得非常广泛。图1展示了一张在一个模具制造公司的工厂中可以找到的典型详细的图纸。在图中表明了孔特征和用来规定这些孔位置的基准尺寸。可以发现这些尺寸显得非常拥挤，并且人工的来确定这些基准尺寸的位置是非常繁重的工作。最终这张满带尺寸的图纸的质量很大程度上依赖于这张图纸绘图员的经验。这项研究的目的是发明一种能够从一个给定的注射模具的零件上自动的产生基准尺寸。尺寸结果必须满足两个要求：第一，任意两个尺寸标记不能够重叠；第二，尺寸标记必须尽可能的接近被测特征。这个问题的关键是研究一种使基准尺寸位置的最优化的方法。2.相关工作在规定机械零件或组装特征的大小或位置信息中尺寸标记和公差分析是两个非常接近的工作，并且大多数的研究工作都集中于公差上。公差的主要研究问题是显示、分析和合成。公差显示与把公差信息合并产品模型草图中相关。实例中包括由Requicha发明的实体偏移法、Turner的可行性空间法和Desrocher和Dlement的TTRS法。在Roy和Yu的文章中有更加详细的观点。公差分析的目的是确定公差中决定零件公差的合并效果。它被用来证实已知或猜想的给定设计单个零件尺寸变化的功能性。公差分析的技术实例包括Monte，Carlo总结和指出的线性方法。它的主要观点是综合合成或公差分配，它是用来确定基于给定装配的功能性要求的零件公差。最近，IsLam同时发表了一个工程方法来解决这个问题。从工程性考虑，根据不同用户需求和技术需求在系统的分析功能性需求基础上，提出了一种解决尺寸需求的方法。FDT软件也是用来支持实现这种方法。FDT向功能需求或尺寸基础中提供用来代表功能需求、尺寸、公差或过程能力的工具。在这个基础上，这种获取的函数方程被分成若干组，并且每组都用来解决所涉及到的函数需求和公差问题的专一方法策略。在公差公差分析和合并中更多详细的观点可以在Roy、Ngoi、Ong，Hong和Chang的文章中找到。从一个CAD模型中自动的产生尺寸的方法已经被提出。在立体几何学实体模型技术上的零件自动显示尺寸上Yuen做过一些早期的尝试。从实体模型上提取一些来自二维表面和剪切剪切圆柱上的点。这些点的坐标被安排在一个树型结构上来产生线性尺寸，并刊登了一个为直径和半径尺寸的简单例子。其他早期的自动尺寸方面的工作由Yu总结了。最近，Chen发表了一个更加在自动尺寸方面具有深度的研究。他们的方法是分析多余尺寸，对称特性的尺寸草图。，选择适当的视角来规定尺寸，并且用一种专业级系统的方法来确定尺寸定位。这个专业级系统分析被测特征的几何形状和布局，并且确定一个合适的位置来放置一个基于一套规则的相关被测特征的尺寸。在一个尺寸位置完成后，构造一个禁区使所有随后的尺寸不会放在此区域中。这就避免了两个尺寸的重叠或交叉。现存的尺寸放置方法是有限的，取决于这个方法的自然连续性。例如，在Chen的方法中，特征的测量是被优先考虑的，但尺寸的位置是随后考虑的。这种方法不适合确定坐标尺寸的位置，特别是在注射模具表面尺寸非常拥挤的情况下。这是因为一个基准尺寸的位置影响到其他尺寸的位置。这篇文章讲述了我们在解决坐标尺寸位置方面的工作。使用了动态程序方法使其最优化，这种新的方法克服了在现存方法中连续方法的限制。3.在基准尺寸中，特征位置通过特征的参考位置和参考基准的水平和垂直距离来确定。基准尺寸的缺省形式在图2a中表示。当被测两特征的垂直距离小于尺寸标记大小（尺寸文本高度的和和相邻两尺寸文本的最小空间）,图2b中的被选形式就需要了。为了防止不重叠，这个尺寸标记则从缺省位置向上或向下转换。就像图2c中，尺寸标记的转换是尺寸的单一延长线打断成了三个部分：被一个倾斜部分相连的两个水平部分。通过三个参数使尺寸标记的范围能够变换调整：（i）倾斜部分和尺寸线的水平部分的折角；（ii）尺寸文本和零件边缘距离m；（iii）特征的位置和最低位置由下式给出： = （1）4自动基准尺寸自动基准尺寸的目的是寻找一种使每个基准尺寸位置都达到最优的方法。每个过程包括两个阶段：准备阶段和最优化阶段。在准备阶段，使最优化过程得到简化的主要参数将被建立。所有特征使用给出的折角、边缘偏移和尺寸标记大小将会进行尺寸位置的可行化测试。在最优化阶段，使用的是动态程序方法。尺寸标记位置最优化是在最重不同的基准系列，包括从他们的缺省位置中每个尺寸转移的最小量，或者从缺省形式使用的尽可能多的最大量。4.1准备阶段被测特征首先被分成一个或更多的特征系列。对于一个特征系列的每一个特征来说，在此特征系列中至少存在另一个特征以便使他们之间的垂直距离小于尺寸标记的大小。换句话说，在一个特征系列中的所有特征，没有重叠的情况下在相邻两尺寸标记不能用缺省的专用形式测量。相反，至多一个特征能用一个叫做尺寸块的特征相关联。尺寸快的构造涉及到每个在尺寸块中的基准尺寸的形式和位置。对于一个尺寸块的每个位置，它的构造是唯一确定的。图3表示在两个构造中的两个特征系列和它们的尺寸块定义1：构造的正确性。假如在一个尺寸块中任意两个尺寸标记间没有重叠，并且每个尺寸标记位于它们的末端位置，则这个尺寸块的构造就是正确的。在图3b中表示的尺寸块就是正确的。图4表示的两个尺寸块是不正确的。因为在图4a中的尺寸标记有重叠所以不正确，图4b的构造中，14.00尺寸标记的延伸线的位置太高，而尺寸要求的位置标记在它之下。定义2：构造末端。有两种构造末端：最高构造和最低构造。假如一个尺寸是正确的，并且任何再高点（或低点）的位置都是不正确的构造则称此尺寸块是最高（或最低）构造。尺寸块的末端构造取决于和。图5a里的构造处于它的最高构造。因为29.5是它的最高位置，它不能再向上移动了。图5b表示的是最低构造，因为14.00是它的最低位置，它不能向下移动了。尺寸块的末端构造有两个在最优化阶段所应用的重要参数。在两个尺寸标记中没有任何重叠的情况下，看测量所有特征是否可行的过程中，这两个参数是非常有用的。在形成一种测量末端构造中可以看到这两个参数是非常有用的。特点1：在一个尺寸标记处于它的最高（或最低）的位置上，最少有一个尺寸标记处于最高（或最低）的位置。特点2：只要尺寸块有末端构造，则它就是不正确的。特点1可以由反证法证明。假如一个尺寸处于它的最高（或最低）构造上，并且没有尺寸标记处于最高（或最低）位置。因此所有它的尺寸标记都不处于它们的最高（或最低）位置，它们能够同时向上（或向下）移动相同距离直到它们中的任意一个达到了最高（或最低）位置上。因为所有的尺寸标记都是同时移动了相同的距离，尺寸标记就不会重叠，并且因此构造结果仍然是正确的，并且在一个比它初始构造更高点（或低点）的位置上。这就违反了初始构造处于最高或最低的假设。特点2可以直接证明。给定的一个正确构造、尺寸块移动更高（或低）直到一个或更高的尺寸标记达到最高（或最低）位置。因此所有尺寸标记都同时移动相同距离，重叠也不会发生。另外，因为至少它的一个尺寸标记达到最高（或最低）位置，尺寸块在没有不正确构造的情况下，尺寸块不能被移动更高（或更低）。根据定义2，构造的结果也是最高（或最低）位置。另一方面，尺寸块是否有末端构造就很明显了，因为末端构造定义是正确的所以该尺寸块就是正确构造。特点1表明尺寸块的末端构造可以由通过观察块中的尺寸标记的末端位置的方法获得。尺寸块的构造可以通过来规定，i=1，2，3，n，是特征系列中第i个特征的尺寸标记的位置。这说明了是按照他们的垂直上升顺序排布的（假如ij，则）。然而，为了避免尺寸标记的重叠，第i个尺寸标记的位置由下式给出： ； ni2 （2）式中SIZE是尺寸标记的大小，是该系列中的第一个特征（）尺寸标记的位置。也同样被用作尺寸块的参考位置。如果构造是正确的，通过式（1）得出所有尺寸标记必须位于其自己最高位置的下端，如下：因此以上关系必须通过所有i证明。因此，最高允许值通过给出： （3）的值通过式（2）给出，一个或更多的都等于。所有其他的都小于它的。没有更大的值因此构造满足，假如正确，这个最终构造就是它的最高构造。然而，在此构造中通过式（2）给出的中的一些值可能小于。因此，每个必须验证。假如所有的i的，则这个最高构造成立。假如一些i的，则这个构造成立就是不正确的，并且没有最高位置成立。通过特性2特征系列不存在任何正确构造。为了寻找最低位置，对于所有i的，最低允许值由下式给出：对于一个零件来说，它的尺寸块是多样化的，假如所有尺寸块中的两个末端构造能够成功建立，则在每个没有任何重叠尺寸块中放置尺寸标记就是可行的。然而，一个尺寸块中的一个尺寸标记可能与另一个尺寸块中的尺寸标记重叠。因此下一步就是测试所有没有任何重叠的尺寸块的放置是否可行。下面所讲的特点3在解释这个测试过程就是十分有用的。特点3：在一个尺寸块中的两个末端构造之间通常建立一个正确的构造。在特点3中很容易发现，从最低位置开始，在尺寸块中的尺寸标记能够同时移动相同的距离，以使它的尺寸标记能够达到它的最高位置。根据定义1，此位置的构造是正确的，当它的一个尺寸标记达到了最高位置，就达到了尺寸块的最高位置。为了测试放置没有重叠的所有尺寸块的可行性，一个特征系列的所有特征首先参考按照它们特征的上升顺序挑选。然后所有特征系列根据它们的第一个特征的位置按照上升顺序排列。第一个尺寸块被放置在它们最低构造的位置。对于第二个特征系列，假如比的顶端高，尺寸块也被放置在它们最低构造的位置，否则在和的范围内放在的顶端。根据特点3，对于后者也是正确构造。否则，作为正确的构造在没有与尺寸块重叠的情况下也不能够建立。对于下个尺寸块来说是一个重复的过程，直到所有正确的尺寸块都已建立，否则无论对于一个尺寸块正确的构造不能建立的话都将中止。4.2最优化阶段在准备阶段之后，每个尺寸块的边缘构造已经建立，并且使在一个尺寸块和两个相邻尺寸块确定避免。使用一种动态工程方法来确定每个尺寸块最优构造。像上面假设一样，根据尺寸块的参考位置用上升顺序来挑选尺寸块。确定尺寸块构造的过程被分为步，i=1，2，3，n，在第i步确定了尺寸块的构造。在每个步，的每个可能构造与其状态相联系。换句话说，状态与第i个尺寸块的第j个构造相联系。选取第步的状态，的费用通过一个总的费用函数来反映，它是通过下式给出：是费用函数用来反映：（i）在独立的状态和的情况在两尺寸块和的相互作用；（ii）尺寸块从它缺省构造中的偏差范围；（iii）两尺寸块和之间的重叠；（iv）尺寸块与被称为禁区的一系列区域之间的重叠。禁区由用户规定，并且禁区通常为由用户放置的其他尺寸标记的区域，因此不允许把尺寸标记放置在远的地方。最优的解决方法从中获得。这个步骤系列=1，2，3，n使最优化得以提高，同时这一系列构造使尺寸标记的放置位置得到最优化。4.3状态解决方法通过特点3，一个尺寸块的两个边缘构造之间的构造是正确的构造，因此每一步都有无数种选择。为了使动态工程方法起作用，必须使用离散化使无数的状态为每个步。离散化最简单的方法是从 和的位置中提取一个合适的数字，这种方法对利用计算机资源来说不是一个有效的方法。这是因为尺寸块的范围（由-给定）可以很大限度的改变，从式1中可以明显的看出尺寸块给出了一个很大的范围，而很少值的给了一个很小的范围。在有合适数量的状态下，那些有较大范围的尺寸块会得到近似的解决方法。另一方面，对于一个有很大范围的尺寸块来说，就像大多数在两边缘构造的正确构造将在它们的尺寸块重叠。通过一个例子最好的解释：假如在尺寸块最高构造的顶端高于尺寸块最高位置构造的底端，则尺寸块在构造上必然与的最高端重叠。相似的问题在最低构造上。因此，对于每个尺寸块来说，一个合适的范围，由和之间的不同来定义：=式中和是尺寸块和中尺寸标记的数量且是独立的。使用这种可行的范围定义，那些在两相邻尺寸块之间经常导致重叠的构造从可行范围内排除。4.4费用函数一个阶段的费用和费用函数都是矢量和矢量值函数。一个费用矢量包括按重要性由高到低排列的5个部分，i=1，2，3，5。动态的实施阶段要求最小的费用选择，因此必须在两矢量间进行对比。两个费用函数通过比较它们的组成来比较。比较从认为最重要的第一部分开始。假如两矢量的第一部分相等，则比较认为是次重要的下一部分。当相关联的两部分不相等时就停止比较。在费用矢量尖的比较是建立在不相等的第一对部件上的。费用函数的第一组成部分等式应该减少相邻两尺寸块和和尺寸块和禁区之间的重叠。由下式给出：式中如果在和之间没有重叠=0。如果有重叠，则就是一个很大的值。费用函数用来减少与状态相关联的尺寸块和禁区之间的重叠。的所有和m认为是与所有禁区相重叠的值。在和如果没有重叠，则=0；如果有重叠，则就是一个很大的值。接下来的四个等式和，是分配四个可选的亚费用函数，这些函数根据四个不同的基准通过返回值来反映构造的。的布局与亚费用函数由用户确定。这就给了用户的可行性来决定准则的重要联系。要求尺寸块放置在特征的中间部分能够被测缺省构造测量的位置，这就认为是尺寸块的缺省构造。亚费用函数是用来测量尺寸块在它的第j个构造处偏移的程度。这个偏移值可以从特征的平均位置上来测量，并且是第j个构造尺寸标记的平均位置=式中为尺寸块中尺寸标记的数量，为在第j个构造上尺寸标记的位置，为特征的位置。和是所有尺寸标记和特征的数量。在只被用作亚费用函数的时候，不能通过费用函数来确定。因为在每两个状态下偏移的数是确定的费用函数是仅建立在能够给出相同费用的可选费用函数上的，可以忽略这个问题。可以分配到另一个可以给出一个较低状态费用函数的部件等式。假如尺寸块在状态时不在它的缺省状态，取1，否则取0。可能要求由用户规定一个可行范围的百分比来限制尺寸块由缺省位置偏移的范围。亚费用函数是用来减少过大的偏移。假如pYFmaxi YFmin ，则的值为1，否则为0。当两个相邻尺寸块在它们的缺省位置重叠，某个值a，则在此重叠中向上移动距离为，并且向下移动到，距离为，其中+=。其中要求=。这就要求两尺寸块移动量相等。亚费用函数不能达到这个目的，因为这仅是测量移动值不是相邻尺寸块移动值的分配。则要求两相邻尺寸块的移动值相等。在两相邻尺寸块移动值的范围通过下式设置不同值：这四个可选的亚费用函数可以由用户自由的选择，安排第二到第五部分来达到目的。在下一部分中，通过例子来说明选择不同的亚费用函数达到不同的结束。5.实例说明合适的最优化方法已经被执行，并且通过API插入到UG2系统中。为了说明亚费用函数的作用，思考图6中的作用。图6中有5个尺寸块，所有尺寸块都处于它们的缺省构造，并且在和之间，和 磨磩涨CAD/CAM涨嵽CAD/CAM 1 CAD/CAMCADCADCAD/CAM鹤涨CADCADchen繤硣 磨磩1涨硣顣硣 2. 涨RequichaTurnerDesrocherDlementTTRSRoyYu仯MonteCarloIs LamFDTFDT硢飬鶼漰RoyNgoiOngHongChang CAD弸Yuen磬緽YuChen緽磬涨磬綨硣粻档 磬Chen淽 3.2a,2b2c任iiimiii = = 1 4 4.1 仰漰3 1 3b44a4b14.00 2 5a29.55b14.00 1 2 1紦衣 2鲻2棬 1涨i=123niiji ni2 2 SIZE 1 i 3 22ii2 i 3 3 31 鱻3鶼 4.2 顣鹹i=123ni仰ij iiiiiiiv涨=123n 4.3 3 -1棬壺 = = 壬侭 4.4 5i=1235 =0m=0 jj = j 10 涨İpYFmaxi ?YFmin 10 a+= 岿 5. APIUG2665飬鶼 DOI 10.1007/s00170-004-2374-2ORIGINAL ARTICLEInt J Adv Manuf Technol (2006) 28: 370378C.L. Li K.M. Yu Y.H. LeeAutomatic datum dimensioning for plastic injection mould designand manufacturingReceived: 7 May 2004 / Accepted: 10 August 2004 / Published online: 20 April 2005 Springer-Verlag London Limited 2005Abstract Datum dimensioning (or ordinate dimensioning) tech-nique is very popular in plastic injection mould drawings wherethe location dimensions of a large number of hole features mustbe specified in the drawings of the mould plates. Although com-mercial CAD/CAM systems provide semi-automatic tools to as-sist the designer in the dimensioning process, it is still a verytedious process, as the user has to specify the location of each di-mension tag. This paper reports a completely automatic methodwhere optimal placements of the dimension tags can be deter-mined. The method employs dynamic programming technique tooptimize the dimension process with respect to several criteriathat can be selected by the user. The method has been imple-mented and incorporated into a commercial CAD/CAM system,and examples are given to illustrate the important features of theprogram.KeywordsAutomatic dimensioning Datum dimensioning Dynamic programming Optimal dimensioning Ordinate dimensioning1 IntroductionCAD/CAM systems are now widely used in the plastic injec-tion mould-making industry. Many companies are using a solidmodeling system to design the injection mould. They use a CADsystem to model not only the core and cavity inserts of the mould(which are the most important components that form the im-pression of the mould), but also all other components in theC.L. Li (u) Y.H. LeeDepartment of Manufacturing Engineering and Engineering Management,City University of Hong Kong,Tat Chee Avenue, Kowloon, Hong KongE-mail: meclli.hkTel.: +8-52-27888432Fax: +8-52-27888423K.M. YuDepartment of Industrial and Systems Engineering,The Hong Kong Polytechnic Universityentire mould assembly. With the advance in Internet technologyand the recent development of Internet-enabled CAD, the de-sign information of the injection mould can be communicatedelectronically between the product engineer (who designs theplastic part) and the tooling engineer (who designs the injectionmould), even though they may be located in different geographicregions of the world. While flow of design information betweenproduct design and tooling design are communicated effectivelythrough an electronic means, the communication of manufac-turing information to the shop floor is done by both electronicand traditional techniques. Computer Numerical Control (CNC)machining toolpath or inspection instructions can be generateddirectly from the CAD/CAM system and downloaded througha network to the CNC controller for the machining or inspectionoperations. However, set-up instructions for a particular machin-ing job may be specified in an engineering drawing. Moreover,not all machining tasks are done using CNCmachine tools. Sometraditional machining processes, such as drilling and grinding,are done using conventional machine tools because of cost con-sideration. Conventional engineering drawings are thus still play-ing an important role in communicating engineering informationto the shop floor. The orthographic projections in engineeringdrawings can be generated automatically from the CAD modelof the parts. Automatic tools for dimensioning of the parts arealso provided by many commercial CAD systems. However, aspointed out by Chen et al. 1, those automatic dimensioningtools are not able to generate dimensions according to the draw-ing standards and engineering practices adopted in the shop floor.In the specific application of injection mould design, datumdimensioning (or ordinate dimensioning) of hole features areused extensively. Figure 1 shows a typical detail drawing that canbe found on the shop floor of a mould making company. Shownin the figure are the hole features and datum dimensions whichare used to specify the locations of the holes. It can be seenthat the dimensions are very crowded and it is a tedious task tomanually adjust the placement of all the datum dimensions. Thequality of the final fully-dimensioned drawing thus depends verymuch on the experience of the draftsman who produces the draw-ing. The purpose of this research is to develop a tool that can371Fig.1. Use of datum dimensioning in a drawing of a plastic injection mould partgenerate the datum dimensions automatically from a given partof the injection mould. The resulting dimensions must satisfytwo obvious requirements: first, that no two dimension tags mayoverlap; and second, that a dimension tag be placed as close aspossible to the feature being dimensioned. The key issue in thisresearch is to develop a method that can optimize the placementof the datum dimensions.2 Related workWhile dimensioning and tolerancing are two closely related pro-cesses in specifying the size and location information of thefeatures in a mechanical part or an assembly, most of the pastresearch work has focused on tolerancing. The major researchissues in tolerancing are representation, analysis and synthesis.Tolerancing representation is concerned with the incorporationof tolerance information into a product modeling scheme. Exam-ples include the solid offset approach developed by Requicha 2,the feasibility space approach proposed by Turner 3, and theTTRS by Desrochers and Clement 4. More detailed review canbe found in Roy et al. 5 and Yu et al. 6. Tolerance analy-sis aims to determine the combined effect of part tolerances onthe assembly tolerance. It can be used to verify the functional-ity of a design given known or assumed variations of individualpart dimensions. Examples of technique in tolerance analysis in-clude Monte Carlo simulation 7 and the direct linearizationmethod 8. The main objective of tolerance synthesis or tol-erance allocation is to allocate part tolerances based on givenfunctional requirements of the assembly. Recently, Islam 9 re-ported a concurrent engineering approach to address this prob-lem. Based on a systemic analysis of the functional requirementsfrom different customer requirements and the technical require-ments from engineering considerations, a methodology for ex-tracting dimensional requirements is developed. A software pro-totype FDT 10 is also developed for supporting the implemen-tation of the methodology. FDT provides tools for representingthe functional requirements, dimensions, tolerances and processcapability into a functional requirement/dimensions matrix. Thefunctional equations captured in the matrix are then separatedinto groups, and each group is then solved using a solution strat-egy specific to the functional requirement and the tolerancingproblem involved. More detailed review intolerance analysis andsynthesis can be found in Roy et al. 5, Ngoi and Ong 11 andHong and Chang 12.Several methods have been developed for generating dimen-sions automatically from the CAD model of a part. Yuen etal. 13 reported an early attempt in automatic dimensioning ofparts represented in Constructive Solid Geometry (CSG) solidmodeling technique. Points from planar faces and axes of cylin-ders are extracted from the solid model. The coordinates of thepoints are arranged in a tree structure to generate linear dimen-sions in the three principal directions. A simple technique fordiametric and radial dimensions was also reported. Other earlyworks in automatic dimensioning have been summarized by Yuet al. 6. Recently, Chen et al. 1,14 reported a more in-depthstudy of automatic dimensioning. Their method analyzed di-mension redundancy, determined dimensioning schemes that arespecific to feature patterns, selected appropriate views for spec-ifying the dimension, and determined the appropriate locationof the dimension using an expert system approach 15. The ex-pert system analyses the geometry and topology of the featureto be dimensioned, and determined a position for placing thedimension based on a set of rules that is relevant to the cur-rent dimensioning feature. With the placement of one dimension,a forbidden region is constructed so that all subsequent dimen-sions will not be placed in this region. This avoids overlap orintersection between two dimensions.372A limitation in the existing approach for the placement ofthe dimension is due to the sequential nature of the method.For example, in Chens 1,14 method the features to be dimen-sioned are prioritized, and the positions of the dimensions aredetermined one after another. The approach is not appropriatefor determining the placement of datum dimensions, especiallywhen the dimensions are very crowded, as in the case of injectionmould plates. This is because the placement of one datum di-mension may have an effect on the placement of another dimen-sion that may be located far away from the current dimension.This paper reports our work in solving the placement problemin datum dimensioning. The major contribution of our work isthe development of a new method that determines the optimalplacement of each datum dimension. Using the dynamic pro-gramming approach to optimization, this new method overcomesthe limitation of the sequential approach used in the existingmethod.3 Basic characteristicof datum dimensioningIn datum dimensioning, the location of a feature is specifiedby the horizontal and vertical distances from the reference lo-cation of the feature and a reference datum. The default formof datum dimension is shown in Fig. 2a. When the vertical dis-tance between two features to be dimensioned is less than thedimension tag size (i.e. the sum of the dimension text heightand the minimum spacing between adjacent dimension texts),Fig.2. Basic characteristics of datum dimensioningthe alternative forms shown in Fig. 2b are required.1The di-mension tags are shifted upward or downward from the defaultlocation to prevent overlap. As shown in Fig. 2c, the shiftingof the dimension tag is achieved by breaking the single exten-sion line of the dimension into three segments: two horizontalsegments which are connected by one inclined segment. The ex-tent to which a dimension tag can be shifted is governed bythree parameters: (i) the dogleg angle , which is the angle be-tween the inclined segment and the horizontal segments of thedimension line; (ii) the margin distance m between the dimen-sion text and the part boundary; and (iii) the location (xfi, yfi) ofthe feature fi. The two extreme positions (i.e. the uppermost pos-ition ymaxiand lowermost position ymini) of the dimension tag aregiven by:ymaxi= yfi+(xfi+m)tanymini= yfi(xfi+m)tan(1)4 Automaticdatum dimensionThe objective of the automatic datum dimensioning system is tofind an optimal position for each datum dimension. The processconsists of two phases of operation: the preparation phase andthe optimization phase. In the preparation phase, major param-eters that facilitate the optimization process will be established.Feasibility for placing the dimensions for all the features usingthe given dogleg angle, margin offset and dimension tag sizewill also be tested. In the optimization phase, a dynamic pro-gramming approach is used. The dimension tag locations can beoptimized with respect to different sets of criteria, including theminimization of the shift of every dimension from their defaultlocations, or maximization of the use of the default form as muchas possible.4.1 The preparation phaseThe features to be dimensioned are first grouped into one ormore feature sets. For each feature in a feature set, there existat least one other feature in the set such that the vertical dis-tance between them is less than the dimension tag size. In otherwords, the features in a feature set cannot be dimensioned usingthe default form exclusively without overlap between adjacentdimension tags. Instead, at most one feature can use the de-fault form while all others require the use of the alternativeform. The set of dimension tags associated with a feature setis called a dimension block. The configuration of a dimensionblock refers to the forms and locations of each datum dimen-sion within the dimension block. For each position of a dimen-sion block, its configuration is uniquely defined. Figure 3 showstwo feature sets and their dimension blocks at two differentconfigurations.1To simplify the explanation of the technique, only vertical dimensionsplaced on the left hand side of the part are discussed. The method developedis general and can be applied to the other sides of the part.373Fig.3. Feature sets and different con-figurations of dimension blocksDefinition 1: Validity of a configuration. A configuration of a di-mension block is valid if there is no overlap between any dimen-sion tags in the dimension block, and each dimension tag lieswithin its extreme positions.The configurations of the dimension blocks shown in Fig. 3bare valid. Two examples of invalid configuration are shown inFig. 4. The configuration shown in Fig. 4a is invalid because twoof the dimension tags overlap. For the configuration shown inFig. 4b, the extension line of the dimension tag 14.00 is at itslowermost position, while the required position for the dimen-sion tag is beyond this lowermost position.Fig.4. Invalid configurations of a dimension blockFig.5. Dimension block at extreme configurationsDefinition 2: Extreme configurations. There are two extremeconfigurations: the uppermost and lowermost configurations.A dimension block is at its uppermost (lowermost) configurationif the dimension block is valid and is at a position such that anyother higher (lower) position results in an invalid configuration.The extreme configurations of a dimension block diare denotedby Ymaxiand Ymini.Figure 5a shows a dimension block at its uppermost config-uration. It cannot move further upward because the dimensiontag 29.5 is at its highest position. Figure 5b shows a dimen-sion block at its lowermost configuration. It cannot move fur-374ther downward because the dimension tag 14.00 is at its lowestposition.The extreme configurations of a dimension block are the twoimportant parameters that will be used by the optimization pro-cess. They are also useful in testing whether it is feasible todimension all the features without any overlap between the di-mension tags. It is observed that two properties are useful indeveloping a method to determine the extreme configurations.Property 1:. For a dimension block at its uppermost (lowermost)configuration, at least one of its dimension tags is at its upper-most (lowermost) position.Property 2:. A dimension block has a valid configuration if andonly if it has extreme configurations.Property 1 can be proved by contradiction. Assume that a di-mension block is at its uppermost (lowermost) configuration, andnone of its dimension tags are at their uppermost (lowermost)positions. Since all the dimension tags are not at their uppermost(lowermost) positions, they can all be moved upwards (down-wards) simultaneously by the same amount until any one of themreaches its uppermost (lowermost) position. As all dimensiontags are moved simultaneously by the same amount, the dimen-sion tags do not overlap, and thus the resulting configurationis still valid and at a higher (lower) position than its originalconfiguration. This violates the assumption that the original con-figuration is the uppermost (lowermost) configuration.Property 2 can be verified directly. Given a valid configura-tion, the dimension block is moved upward (downward) until oneor more of its dimension tags reach its uppermost (lowermost)position. Since all the dimension tags are moved simultaneouslyby the same amount, overlap does not occur. Moreover, the di-mension block cannot be moved upwards (downwards) any fur-ther without invalidating the configuration because at least oneof its dimension tags is at its uppermost (lowermost) position.According to Definition 2, the resulting configuration is thus theuppermost (lowermost) configuration. On the other hand, it isob-vious that if a dimension block has extreme configurations, thenit has a valid configuration because the extreme configurationsare, by definition, valid.Property 1 indicates that the extreme configurations of a di-mension block can be obtained by investigating the extreme pos-itions of the dimension tags in the block. The configuration ofa dimension block can be specified by yi,i =1,2,.,n, whereyiis the location of the dimension tag of the ith feature in thefeature set fi. This assumes that fi are arranged in ascend-ing order by their vertical positions (i.e. yfi yfjif i j). Then,to avoid overlap between dimension tags, the location of the ithdimension tag is given by:yi= (i 1)SIZE+ y1;n i 2(2)where SIZE is the dimension tag size and y1is the location of thedimension tag for the first feature ( f1) of the set. y1is also usedas the reference location of the dimension block.For a configuration to be valid, all dimension tags must liebelow its own uppermost position given by Eq. 1. That is:ymaxi yiand thusymaxi (i 1)SIZE + y1The above relationship must be satisfied by all i. Therefore, thehighest allowable value for y1is given by:Miniymaxi(i 1)SIZE(3)with the y1value given by Eq. 2, and one or more yiequal toymaxi. All other yiare less than its ymaxi. Since no other largervalue of y1results in a configuration that satisfies ymaxi yi, theresulting configuration, if valid, is the uppermost configuration.However, it is possible that at this configuration some of the yigiven by Eq. 2 is less than ymini. Therefore, a check is performedfor each yi. If yi yminifor all i, then the uppermost configura-tion is found. If yii(YmaxjSIZEni)YFmini= Max(Ymini, Maxij1(Yminj+SIZEnj)where niand njare the number of dimension tags in dimensionblocks diand dj, respectively. Using this definition of the feas-ible range, those configurations that always cause overlap withadjacent dimension blocks are excluded from the feasible range.A fixed resolution, say 0.5mm, is specified and the number ofstates for a given stage is obtained by dividing the feasible rangeby the given resolution.4.4 Cost functionsThe overall cost of a stage and the cost function Ci(ti,j,ti1,k)are vector and vector-valued functions, respectively. A cost vec-tor consists of five components ci, i = 1,. ,5 arranged in de-scending order of importance. That is, ciis considered moreimportant than cjifi p?YFmaxiYFmini?, and is set to zerootherwise.When two adjacent dimension blocks at their default loca-tions overlap for a certain amount a, the overlap can be removedby shifting diupwards by ai, and shifting di1downwards byai1, such that ai+ai1= a. It may be desirable that ai= ai1.That is, the required total shift is being shared equally betweentwo dimension blocks. The sub-cost function DV(ti,j) is not ableto achieve this purpose because it only measures the total shiftamount and not the distribution of the amount between adjacentdimension blocks. DE(ti,j) is devised to equalize the amount ofshift between adjacent dimension blocks. It is set to the differ-ence between the extent of deviations of the adjacent dimensionblocks.DE(ti,j,ti1,k) = |DV(ti,j) DV(ti1,k)|.The four optional sub-cost functions are freely selected andassigned to the second to fifth cost components by the user toachieve different objectives. In the next section, examples aregiven to illustrate the different results obtained by choosing dif-ferent optional sub-cost functions.5 Illustrative examplesThe proposed optimization method has been implemented usingC+ and incorporated into the Unigraphics II CAD/CAMsystemthrough its Application Program Interface (API). To illustrate theeffects of the sub-cost functions, consider the example illustratedin Fig. 6. There are five dimension blocks in the figure. All thedimension blocks are at their default configurations and overlapoccurs between d1and d2, and between d4and d5. These dimen-sions are generated automatically by the optimization program,with the cost function contributed only by the sub-cost functionDV. As a result, the dynamic programming procedure selectsa set of states where the lowest cost corresponds to the minimumdeviation from the default location of each dimension block.Fig.6. Dimension blocks at their default configurationFig.7. Minimizing the amount of shift and the number of dimension blocksto be shifted to remove overlaps between dimension blocks377Fig.8. Comparison of the effects obtained from DNand DEFig.9. Execution of the optimization program to gen-erate datum dimensions for an example part378By activating the cost component equation c1which dealswith overlaps, and setting cost component c2to DVand c3toDN, the optimization program shifts the dimension blocks fromtheir default configurations by a minimum amount to removethe overlaps, and also the number of dimension blocks that haveto be shifted are also minimized. The resulting dimensions areshown in Fig. 7. Only two dimension blocks d1and d4are shifteddownwards to remove overlaps with d2and d5.Figure 8 shows another example which compares the effectof DNand DE. Figure 8a shows the result obtained by using DVand DN. The lower block is shifted while the upper block re-mains at its default configuration. Figure 8b shows the result ob-tained by using DVand DE. Both dimension blocks are shiftedby a small amount to remove the overlap.Figure 9 shows the execution of the optimization program inthe Unigraphics II CAD/CAM system. After invoking the userfunction that contains the optimization program, a sequence ofdialogue boxes appear to prompt user input. The user selects theface of a part where datum dimensions are to be generated andthe reference datum to be used through the first and second di-alogue boxes, respectively. Finally, the user is prompted for thefont size of the dimension text and the margin offset to be used.A drawing that contains 120 datum dimensions for all hole fea-tures associated with the selected face is generated automaticallywithin a second, as shown in the figure. Several typical exampledrawings in injection mould design with 50 to 250 hole featureshave been used to test the performance of the program. The testsshow that datum dimensions can be generated within a second,which confirms that the performance of the program is suitablefor interactive use.A limitation of the current implementation is that the dog-leg angle is set to a fixed value, and cannot be adjusted by theoptimization program. When it is not feasible to place all thedatum dimensions, the progr