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化工搅拌器的设计 化工 搅拌器 设计

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黄河科技学院本科毕业设计（论文）任务书 工 学院 机 械 系 机械设计制造及其自动化 专业 2010 级 专升本 班学号 学生 指导教师 毕业设计（论文）题目 化工搅拌器的设计 毕业设计（论文）工作内容与基本要求（目标、任务、途径、方法，应掌握的原始资料（数据）、参考资料（文献）以及设计技术要求、注意事项等）一、设计技术要求、原始资料（数据）、参考资料（文献）搅拌器是化工生产中经常使用的设备，该设备可以代替手动搅拌对人体有毒或对皮肤有伤害的化工原料，结构简单，使用方便，在化工生产应用比较广泛。本课题要求设计一个化工搅拌器，容积在600升左右，工作平稳灵活，使用方便。在做本课题时，需要查阅机械制图、机械设计、化工过程与设备等资料。二、设计目标与任务设计出满足要求的搅拌器，并完成该搅拌器的装配图与部分零件图，查阅文献资料不少于12篇，其中外文资料不少于2篇。 1、文献综述一篇，不少于3000字，与专业相关的英文翻译一篇，不少于3000汉字。 2、毕业设计说明书一份，内容与字数都不少于规定的任务量。3、图纸若干（折合后不少于A1图纸3张，可以用计算机绘图）。 4、包含本次设计的所有内容的光盘一张。毕业设计（论文）撰写规范及有关要求，请查阅黄河科技学院本科毕业设计（论文）指导手册。三、时间安排1-4 周 完成开题报告、文献翻译、文献综述及总体方案设计5-10 周 完成总体设计、完成部分机构的装配图及部分零件图并撰写说明书10-12 周 修改论文、资格审查等12 周 毕业答辩毕业设计（论文）时间： 2012 年 2 月 13 日至 2010 年 5 月 15 日计 划 答 辩 时 间： 2012 年 5 月 19 日专业（教研室）审批意见：审批人签名：黄河科技学院毕业设计（论文）开题报告表课题名称化工搅拌器的设计课题来源教师拟订课题类型AY指导教师学生姓名专 业机械设计制造及其自动化学 号一、调研资料的准备本课题是设计类的题型，在做本课题时，已查阅了机械制图、机械设计、机电一体化等课程，以及查阅与课题相关的文献资料。二、设计的目的与要求大学毕业设计，是提高学生综合应用所学的理论知识来处理实际问题的能力，是培养学生理论与实践相结合的一个重要的实践性环节，是对大学四年所学知识总结与运用。因此，本环节在教学过程中有着特别重要的意义。本课题是在此基础上拟定而成的。本课题的主要内容是对化工搅拌器进行设计，该设备是化工生产中经常使用的设备，可以代替手动搅拌对人体有毒或对皮肤有伤害的化工原料，机构简单，使用方便，在化工生产应用比较广泛。本课题设计的化工搅拌器，容积在600升左右。三、设计的思路与预期成果1、设计思路： 根据任务书中的要求设计出化工搅拌器。（1） 总体设计； （2） 搅拌机的内部设计； （3） 电动机及减速器的选型；（4） 支撑装置的设计；（5） 轴的密封；2、预期的成果（1）完成文献综述一篇，不少与3000字，与专业相关的英文翻译一篇，不少于3000字（2）完成内容与字数都不少于规定量的毕业设计说明书一份（3）绘制装配图，部分零件图（4）刻录包含本次设计的所有内容的光盘一张四、任务完成的阶段内容及时间安排 1周2周 收集设计资料并完成开题报告 3周4周 完成英文资料翻译并写出文献综述 5周7周 进行总体设计和部分零部件的选择与设计 8周12周 绘制装配图和部分零件图、编写毕业设计说明书 13周 修改整理，准备答辩五、完成设计（论文）所具备的条件因素 1.修完机械设计、机械制图、机电一体化设计基础等课程，获得一定的理论知识及设计水平； 2.借助图书馆的相关文献资料，以及相关的网络等资源； 3.有多年教学和生产实践经验的导师的指导。指导教师签名: 日期: 课题来源：（1）教师拟订；（2）学生建议；（3）企业和社会征集；（4）科研单位提供课题类型：（1）A工程设计（艺术设计）；B技术开发；C软件工程；D理论研究；E调查报告 （2）X真实课题；Y模拟课题；Z虚拟课题 要求（1）、（2）均要填，如AY、BX等。目 录 1任务书1 2开题报告2 3指导教师评阅表4 4主审教师评审表5 5毕业设计(论文)答辩评审与总成绩评定表66毕业设计说明书77文献综述45 8文献翻译52 9光盘 10设计图纸或实验数据记录黄河科技学院毕业设计（文献综述） 第 6 页 搅拌器的研究与分析摘要： 搅拌机式搅拌设备的心脏。在搅拌机设计及使用过程中，合理的选取搅拌机的结构，运动和工作参数，直接关系到混泥土等材料的搅拌质量和搅拌效率。论文对搅拌臂的排列、搅拌叶片的安装角、拌筒长宽比、搅拌机转速和搅拌时间等主要参数的选取进行分析与实验研究。通过归纳，给出了双卧轴搅拌机的主要参数，包括搅拌臂排列、叶片安装角、拌筒长宽比、搅拌线速度等；给出了评价搅拌机参数合理与否的准则；给出了搅拌臂排列的基本原则。关键词：拌臂排列，叶片安装角，拌筒长宽比，搅拌线速度1 搅拌机的简介通常搅拌装置由作为原动机的马达(电动、风动或液压)，减速机与其输出轴相连的搅拌抽，和安装在搅拌轴上的叶轮组成 减速机体通过一个支架或底板与搅拌容器相连。当容器内部有压力时，搅拌轴穿过底板进入容器时应有一个密封装置，常用填料密封或机械密封。通常马达与密封均外购，研究的重点是叶轮。叶轮的搅拌作用表现为“泵送”和 涡流”，即产生流体速度和流体剪切，前者导至全容器中的回流，介质易位，防止固体的沉淀并产生对换热热管束 (如果有)的冲刷；剪切是一种大回流中的微混合，可以打碎气泡或不可溶的液滴，造成“均匀”。气体和低黏度液体混合机械的特点是结构简单，且无转动部件，维护检修量小，能耗低。这类混合机械又分为气流搅拌、管道混合、射流混合和强制循环混合等四种。中、高黏度液体和膏状物的混合机械，一般具有强的剪切作用；热塑性的物料混合机主要用于热塑性物料(如橡胶和塑料)与添加剂混合；粉状、粒状固体物料混合机械多为间歇操作，也包括兼有混合和研磨作用的机械，如轮辗机等。混合时要求所有参与混合的物料均匀分布。混合的程度分为理想混合、随机混合和完全不相混三种状态。各种物料在混合机械中的混合程度，取决于待混物料的比例、物理状态和特性，以及所用混合机械的类型和混合操作持续的时间等因素。液体的混合主要靠机械搅拌器、气流和待混液体的射流等，使待混物料受到搅动，以达到均匀混合。搅动引起部分液体流动，流动液体又推动其周围的液体，结果在溶器内形成循环液流，由此产生的液体之间的扩散称为主体对流扩散。当搅动引起的液体流动速度很高时，在高速液流与周围低速液流之间的界面上出现剪切作用，从而产生大量的局部性漩涡。这些漩涡迅速向四周扩散，又把更多的液体卷进漩涡中来，在小范围内形成的紊乱对流扩散称为涡流扩散。机械搅拌器的运动部件在旋转时也会对液体产生剪切作用，液体在流经器壁和安装在容器内的各种固定构件时，也要受到剪切作用，这些剪切作用都会引起许多局部涡流扩散。搅拌引起的主体对流扩散和涡流扩散，增加了不同液体间分子扩散的表面积减少了扩散距离，从而缩短了分子扩散的时间。若待混液体的粘度不高，可以在不长的搅拌时间内达到随机混合的状态；若粘度较高，则需较长的混合时间。对于密度、成分不同、互不相溶的液体，搅拌产生的剪切作用和强烈的湍动将密度大的液体撕碎成小液滴并使其均匀地分散到主液体中。搅拌产生的液体流动速度必须大于液滴的沉降速度。少量不溶解的粉状固体与液体的混合机理，与密度成分不同，互不相溶的液体的混合机理相同，只是搅拌不能改变粉状固体的粒度。若混合前固体颗粒不能使其沉降速度小于液体的流动速度，无论采用何种搅拌方式都形不成均匀的悬浮液。不同膏状物的混合主要是将待混物料反复分割并使其受到压、辗、挤等动作所产生的强剪切作用，随后又经反复合并、捏合，最后达到所要求的混合程度。这种混合很难达到理想混合，仅能达到随机混合。粉状固体与少量液体混合后为膏状物，其混合机理与膏状物料混合的机理相同。不同的热塑性物料以及热塑性物料与少量粉状固体的混合，需要依靠强剪切作用，反复地揉搓和捏合，才能达到随机混合。 2 搅拌机的发展史及现状 搅拌混合设备是一种应用广泛、品种繁多的流体机械产品，适用于化工、冶金、医药、食品和饲料等领域。搅拌操作是工业反应过程的重要环节，它的原理涉及流体力学、传热、传质及化学反应等多种过程，而搅拌器是为了使搅拌介质获得适宜的流动场而向其输入机械能量的装置。因此搅拌器也叫做Mixer，或叫做Agitator,Stirrer。广义的搅拌还包括将固体微粒分散悬浮在溶液里面或将溶液变成均匀的乳化液，因此它包括分散器和均质机。某些搅拌器能产生极大的剪切力，以获得细化的粒子比胶体磨大10倍以上的亚微米悬浮体，因此，可用于制造色拉酱、美容乳之类的精细食品和化学品。石化工业常用于聚氯乙烯合金、顺丁橡胶合釜、反应釜、汽提釜等统称为搅拌容器（Agitatored Vessels，或Stirred Vessels）。近年来，搅拌器和搅拌容器获得飞速发展的同时，正面临着满足合理利用资源、节能降耗和对环境保护要求的严峻挑战。搅拌器和搅拌容器在服从装置规模经济化和品种多样化的同时，正日趋大型化。日立制作所自1949年生产搅拌反应釜以来已为聚氯乙烯、对苯二甲酸、苯乙烯单体、聚丙烯等装置生产了搅拌反应釜近4000台，容器的最大容量达576m ，最大直径达7620 mm，圆筒部分最大长度达 44380 mm，设计压力最大 28 MPa，设计温度最高 530 cI二，电机最大功率达 1100 kW。基于节能的要求，开发出变频调速电机、小剪切阻力桨叶、以新型密封代替机械密封和填料密封，以磁力驱动代替机械传动。基于降低产品总体成本、减少维修保养成本和提高设备平均维修间隔时间的要求，大大提高了设备运行寿命。基于满足卫生和降低清洗和杀菌成本的要求，实现了CIP(就地清洗 )和 SIP(就地杀菌)，提高了自动化水平，避免了人与产品的接触，减少了人工操作和待机时间，大大提高了产品的卫生水平。3 搅拌过程及搅拌桨叶的分类搅拌技术观点看，流体搅拌可分为五种基本搅拌应用，而每一种搅拌应用又可根据物理过程和化学过程分为两种类型。因此，总共有十种基本的搅拌应用。每一种基本搅拌应用都有各自的搅拌特点，过程要求和放大设计准则。实际应用时，每种搅拌应用往往会有几种基本搅拌应用组成，如絮凝搅拌过程由液液混合和固体悬浮两个基本搅拌应用组成。搅拌机主要有电机、减速装置、搅拌轴和桨叶等组成。搅拌桨叶的形式多种多样但无论何种桨叶形式，搅拌机在操作时，其轴功率消耗都产生两部分作用，一部分是桨叶产生的排液量，另一部分是桨叶产生的压头。桨叶产生的压头又可分成两部分，即静压头和剪切力；搅拌机桨叶在操作时，必须克服静压头，而剪切力使得物料分散、混合。因此，根据桨叶产生排液量，克服静压头和产生剪切力能力的大小，可将所有桨叶分成三种基本类型，即流动型、压头型和剪切型。每一种桨叶在提供某种基本作用的同时（如流动型桨叶的基本作用是产生排液量），也提供另外两种作用（产生剪切和克服静压头）。 根据不同的搅拌工程对搅拌要求的不同，选择一种合理的桨叶形式，使得搅拌桨叶提供的排液量，静压头和剪切之匹配能最大限度地满足搅拌过程的搅拌要求。如固体悬浮及互容液体的混合，要求桨叶能提供大排液量、低剪切。而气一液分散，要求桨叶能同时提供剪切、排液量和静压。 搅拌桨叶的分类，也可以按照桨叶对流体作用所产生的流动型态来分，可将桨叶分成两种类型-轴流式桨叶及径流式桨叶。所谓轴流式桨叶，是指桨叶的主要排液方向与搅拌轴平行，螺旋推进式桨叶即是一种典型的轴流式桨叶；所谓径流式桨叶，是指桨叶的主要排液方向与搅拌轴垂直。带有“Sabre形状叶片的搅拌桨，搅拌能耗量小，产生的流动为主导轴向型，确保非常有效。带有450倾斜平板叶片的轴向搅拌桨，对中小体积的搅拌最为经济。这种搅拌桨叶产生的流动为主导轴向型带径向流，产生剪切扰动。在不粘的介质中这种搅拌桨叶对大多数应用均非常理想，特别是那些需要高速低能耗的场合。例如： 被用于进行悬浮或热交换。倾斜的桨叶低速运转，产生较高的扰动。这种基本搅拌桨叶通常对一些简单搅拌应用有效。螺旋推进式型桨叶，对小体积的搅拌最为经济。在无粘性的介质中，适合于气-液交换及热交换。用于固体、混合物、乳液的传统桨叶，产生中等水平产生径向流，具高抗动性和高能耗，专用于特殊应用。由于重量原因，这种桨叶仅用小直径，经常用高速运行（电机直接驱动）。4 搅拌机的分类 搅拌机是以混合、揉和方式调整物料稠度的一种机械设备。搅拌机在工业生产中，特别是在建筑、水泥等领域有着非常重要的应用。搅拌机按照的分类方式很多，下分多个种类，以下是常见的搅拌机划分方法与搅拌机种类。4.1 搅拌机的作业方式分类搅拌机按照作业方式上的差别，可以分为循环作业式搅拌机和连续作业式搅拌机两种。循环作业式搅拌机是以周期循环方式，顺序完成供料、搅拌和卸料三道工序，对于物料用量的控制较为精准，物料搅拌的效果较好。目前，在实际生产中应用的搅拌机多属于循环作业式搅拌机。连续作业式搅拌机对物料的处理，同样经过供料、搅拌和卸料三道工序，但是这三道工序是在搅拌机附属的筒体内连续完成的。连续作业式搅拌机对物料的配比控制能力较差、也不易掌握物料搅拌的时间，但连续作业式搅拌机的生产能力较高、生产量较大，适合物料处理效果要求低的搅拌工作。4.2 搅拌机的搅拌方式分类搅拌机按照搅拌方式上的差别，可以分为自落式搅拌机和强制式搅拌机两种。自落式搅拌机是搅拌鼓转动而搅拌鼓内的叶片相对静止。自落式搅拌机工作时，搅拌鼓会旋转带动混合物料，叶片将混合物料提升到一定高度后，物料会在自身的重力作用下洒落，完成搅拌的过程。强制式搅拌机是搅拌鼓保持静止而叶片强制搅拌。强制式搅拌机工作时叶片会在转轴的带动下转动，强制搅拌混合物料。强制式搅拌机的搅拌质量好、搅拌效率高，但是叶片磨损速度很快，且需要很大的动力输出。4.3 搅拌机的装置方式分类搅拌机按照装置方式分为固定式搅拌机和移动式搅拌机两种。固定式搅拌机安装在固定基座上，整机无法移动，生产效率高，多适用于搅拌楼或搅拌站使用。移动式搅拌机安装在汽车上，易于移动、机动性好，多适用于各种小型工程。4.4 搅拌机的容量大小分类搅拌机的设计容量范围很大，从50L到3000L都有。小型搅拌机的出料容量为50到250L，而中型搅拌机的出料容量就上升至300到500L，大型搅拌机的容量则高达1000到3000L。4.5 搅拌机的内部构造分类桨式搅拌器 有平桨式和斜桨式两种。平桨式搅拌器由两片平直桨叶构成。桨叶直径与高度之比为 410,圆周速度为1.53m/s，所产生的径向液流速度较小。斜桨式搅拌器的两叶相反折转45或60，因而产生轴向液流。桨式搅拌器结构简单，常用于低粘度液体的混合以及固体微粒的溶解和悬浮。 5 搅拌机的应用范围 新型搅拌器系换代产品，是化工和建材行业搅拌设备无可替代的产物，实现了正确“搅和拌”的问世，从而淘汰其它搅拌设备所以承但的重任。它以其超常规的构思和精锐的技术含量，合理的设计水准，填补了国际空白。其广泛用于油漆、涂料、染料、制革、医药、饮料、粘胶剂、食品、洗涤品、化妆品及各种固态物体等。有取之不尽的财富。对物体分散、乳化、均质、调色等较之传统搅拌机的搅拌效果更加理想、直观、是搅拌行业的一次革命。另一方面，我们和一些发达国家还存在一定的距离，这就需要我们汲取和借鉴国外的先进技术，使我们的产品更加完美。参考文献1 黎明，化工行业标准-搅拌器M.北京：化学工业出版社,2008-10.2 陈志平，搅拌与混合设备设计选用手册M. 北京：化学工业出版社. 3 李国刚，固体废物实验与监测分析方法M. 北京：化学工业出版社.4 成大先, 机械设计手册M（第五版）（第1卷）. 北京：化学工业出版社.5 初志，吴岩石等编.化工容器技术问答-化工设备技术问答丛书M. 北京：化学工业出版社.6 谭天恩，窦梅，周明华.化工原理M (上)(普通高等教育十五国家级规划教材). 北京： 化学工业出版社,2006年08月7 曲文海，朱有庭.于浦义化工设备设计手册M(上 下). 北京：化学工业出版社.8 倫世儀.生化工程M（第二版）.化学工业出版社.9 汤善甫，朱思明主编.化工设备机械基础M（第二版）.上海：华东理工大学出版社，2004.12. 10Yang D Y, Jung D W, Song I S, etal. 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Development and application of sheet-forming simulation J.Jounral of Materials Processing technology.1995(50)1-16 毕业设计文献翻译 院（系）名称工学院机械系 专业名称机械设计制造及其自动化 学生姓名 指导教师2012年 03月 27日 毕业设计 文献综述 院（系）名称工学院机械系 专业名称机械设计制造及其自动化 学生姓名 指导教师2010年 03月 27日 黄河科技学院毕业设计(文献翻译) 第 7 页基于机床混合模型的参数曲线高速插补速度极值分析 塞巴斯蒂安四蒂马尔，日达吨法鲁克美国加州大学戴维斯分校，机械系和航空工程系，美国 加州956162005年7月7日收稿， 2006年3月23修订 ,2006年4月10日发表摘 要算法是随着估算进给速度的曲率的变化而发展的，这确保了一个3轴的最低运动时间，数控机床受固定轴加速度范围和驱动电机输出扭矩特性的轴速度约束。对于由一个多项式参数曲线指定一个路径，最优时间的进给速度确定一个分段曲线函数的参数解析与细分，对应限制一个轴的加速度饱和常数。进给速度之间的始发点段，可通过数值计算的解决方法。对于细分固定加速度的（平方）的最佳进给速度是合理的曲线参数。对于速度依赖加速度范围，最佳进给速度在一种新的超越函数，其值及封闭的形式表达可有效地计算使用，实时控制一个特殊的算法。最佳进给速度推导出一个实时插补算法，可以直接从驱动器的解析路径描述机器。从实验结果的执行情况看，时间最优的3轴数控由于采用开放式构架的软件驱动进给速度控制器给出。该算法是一种显著的改善蒂马尔支持SD，法鲁克逆转录，史密斯给付，博亚杰夫建议。算法的时间最优控制沿着弯曲的数控机床刀具路径。机器人集成制造2005; 21:37-53，因为除了电压限制运动排除了沿直线或接近直线路径段任意高速的可能性。2006爱思唯尔版权所有。关键词：3轴加工，进给速度的函数，加速度的极值，时间最优路径遍历，噪音控制1 简介时间最优控制在以往的研究领域，机器人技术1-7和数控加工8-10关注与一个指定的路径最短时间穿越了一系统具有已知的动态和在指定的范围运动的执行机构。该方案解决这些问题的一个典型招致控制“噪音”战略，其中至少有一个输出系统饱和执行器在每个瞬间整个路径遍历。这些研究通常假定驱动器常与对称力极限（独立驱动的速度和方向）而且一般不解决问题的速度，超过该范围执行器可以发挥最大的力量。固定场直流电动机是最常见的定位在机器人及数控加工轮廓的应用11。由于他们的扭矩输出是成正比对电枢电流，恒转矩对称限制反映了马达的最大电流容量电枢绕组。保持恒转矩输出不断变化的有关电枢电压反电动势的（正比于电机转速）否则控制电枢电流供应10。除了电枢，电流限制应用电枢电压可能会受到限制的问题引起的电机特性或电枢电源。 这样电压限制限制了生产的运动能力最大输出扭矩，速度有限的范围内。超出此范围，最大适用电枢电压不电枢电流是限制因子电机扭矩输出，速度依赖造成最大力矩电机的增加呈线性下降速度10。 在3轴加工中，最大电流容量一轴驱动电机施加一个恒定的加速度限制在轴速度降低，最大电压容量规定在较高轴速度依赖加速度极限速度。从目前有限的过渡到电机轴的操作发生在过渡速度。在下面的速度过渡的速度，最高轴加速度保持不变。在速度大于过渡的速度，最大轴加速度线性轴的速度下降，在下降到零轴空载速度。为了保证时间的最优路径遍历符合这两个驱动器电流和电压的限制，算法必须考虑到这两个常数和在每台机器轴加速度限制。这本文推广了以前的研究结果9用人唯一不变的加速度式（1假设高速任意核算结果，如果路径中包含扩展线性段），并介绍了新算法现实的时间来计算最优进给速度为笛卡尔与驱动电机轴数控机床同时受电压和电流限制。列入的加速式招致重大，定性以较早的算法在许多方面的变化9，其中包括一套可行的进给速度和加速组合的速度限制曲线（可变编码）;可能的切换不同类型点;以及进给速度的极值函数的形式相平面轨迹。然而，对于笛卡尔数控与轴独立驱动的机器，它仍然是可能的以获取基本上封闭形式解的进给速度，由于计算能力的根源某些多项式方程。我们首先回顾了第2个DC电机运行并在第3轴加速度范围。我们介绍了最低时的遍历问题常和速度依赖轴弯曲的路径加速度限制在第4节，我们得出进给速度恒和速度的表达式依赖极值加速度轨迹。饲料加速度限制，可变长编码，和进给速度破发点，然后对第5-7分别进行讨论。经过讨论的进给速度计算在第8和实时数控插补算法在第9，我们目前的细节进给速度计算和机实施效果。在第10条的几个例子。最后，第11节总结我们的结果并提出了一些结论说这番话的。2 直流电动机转矩限制为加深对轴的性质背景，适当的直角式加速度数控机床，我们开始与一固定场区的简要概述了通常用于驱动小型至中型电机。铣床（见其更完整的细节操作 10）。该方程管运作电机是也就是说，电机的输出转矩T是成正比的，电枢电流I，反电动势是成正比。电机角速度，电枢和应用电压V等于反电动势和总结的压降电枢电阻R的KT和柯相称因素，所谓的扭矩常数和反电动势常数，是内在的物理一个给定的电机性能的影响。从这些表现形式，你可以很容易地推导出电动机转矩转速的关系在给付是失速扭矩,和无负载速度。所以，电机转矩降低,线性电动机的速度增加，从时到时。 参见12更完整的细节。在发动机启动和低速时，反电动势E是小相比，施加电压V，以及限流设备是用来限制电流I为（大约）常数的最大值，以防止伊利姆电枢绕组的损坏。因此，电机转矩输出保持恒定在整个低转速范围的操作。随着马达的加快，电枢电压应用最终达到最大电机或电源供应器额定电压。这发生在过渡的速度，定义对于速度高于催产素大，电枢电压（而不是比目前的）是在电机转矩限制因素输出。在电压限制，扭矩T线性下降随着电机转速澳，下降至零，空载转速的实现。图（1）描述了电机的制约电流和电压范围，和在为积极和消极的马达速度。该约束定义两个平行带，其交集形成定义可行的制度直流电动机运行。所有受理的组合电动机的扭矩和速度，按照给定的电枢电流和电压范围，在这个谎言。 对超出的部分延伸无负载在每个方向符合再生电机，制动其中意味着外部扭矩申请。由于没有这样的扭矩可在驱动器中的数控机床马达，可行的扭矩范围/速度降低状态来表示空载速度最高电机转速，高产的六面平行四边形，如图1所示。 这六个面平行四边形定义了三个不同的直流马达转速范围，具有鲜明的最低和每最大扭矩限制，即：3 轴加速度限制在高速加工8,13,14惯性力可能称霸切削力，摩擦等，尤其是工具路径的高曲率。会计轴惯性，轴的速度和加速度是成比例的力矩电机和电机速度分别。考虑，也就是说，x轴。如果它是有效质量的Mx和驱动，由驱动电机通过弹性模量Kx（即滚珠丝杆，线性轴速度是关系到汽车的角相应的轴加速度以电动机转矩T是ax=KxT/Mx。注意到进给速度可被视为一个数量级v和载体由单位路径切线的特定方向，我们有和电机转速为因此，上面导出的转矩限制相当于X轴加速度限制其中VT是轴过渡的速度，V0的是轴空载速度，我们定义通过对速度的依赖加速度限制，轴速度VX始终保持在区间轴转速范围内 ,最低轴加速度和最高限额都是固定的，因此，这被称为制度的不断限制在X轴。轴速度范围，为其中一个加速度是固定的二是依靠速度，被称为混合为X轴的限制制度。在制度不变的限制，加速范围可写为。对于混合限制制度，加速范围可能表现在表格在路径遍历，每个轴在一个月内运作，其加速度限制制度独立于其他轴，每一个都可能加速极限之间切换，按照制度与工具的变化路径几何形状和进给速度。因此，有四个加速度限制制度的可能组合，其中的x，y轴，Z轴（见表1）。对于一个平面曲线，涉及的仅有的两个机轴运动，有三个可能的组合：常量/恒，恒/混合，和混合/混合。每个组合的加速度极限，除了要具体分析计算的时间最优进给速度。4 时间最优的进给速度考虑到学位曲线描述的路径与对照点。如果指弧长沿曲线测量，我们定义参数速度切线的单位和（主轴）和正常向量曲率（4）定义与此相反，与我们可以写现在假设我们遍历与进给速度（速度的曲线）指定由该函数。由于衍生金融工具方面时间t和参数x，我们以点表示和素数，分别为，由有关速度和加速度向量由每个点给出由切向分量的消失如果V 是常数，而正常（向心力）组件的如果消失K=0。其时的进给速度（衍生的加速度）给出的角度来看，我们希望尽量减少沿线rx遍历时间，开始和结束休息时，受限制的加速度表格（3）和其他类似用语机轴。这些要求可以在以下方面措辞以下优化问题使得其中指的是笛卡尔每一个组成部分，正如在第3节，轴加速度的形式是4.1 恒定加速度轨迹从关系和我们可以写对于给定的曲线的X轴组件（说）一个定义为加速的在我们写，因为它是方便工作对进给速度平方（见9详情）。在一个不断加速阶段极值加速度限制，其中一个组成部分，是加速等于加上或减去相应的约束，一条件是产生一个为q的线性微分方程如果x是加快轴，这个方程承认为（平方）进给速度，即封闭形式解。定义为其中积分常数C是取决于指定的一个已知点，对轨迹：关于进一步解决（10）的方法详情中可以 在9 找到。4.2 加速度极值轨迹考虑到当x轴（假定）执行一个加速极值，通过定义加速度极值约束决定进给速度v形式。通过以上描述，就是在这种情况下推导出进给速度的微分方程在我们不断介绍方程（11）是一阶变系数非线性微分方程。这对来说，可以专门写作Robotics and Computer-Integrated Manufacturing 23 (2007) 563579Time-optimal traversal of curved paths by Cartesian CNC machinesunder both constant and speed-dependent axis acceleration boundsSebastian D. Timar, Rida T. Farouki?Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USAReceived 7 July 2005; received in revised form 23 March 2006; accepted 10 April 2006AbstractAlgorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axisCNC machine subject to both fixed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of theaxis drive motors. For a path specified by a polynomial parametric curve, the time-optimal feedrate is determined as a piecewise-analyticfunction of the curve parameter, with segments that correspond to saturation of the acceleration along one axis under constant or speed-dependent limits. Break points between the feedrate segments may be computed by numerical root-solving methods. For segments thatcorrespond to fixed acceleration bounds, the (squared) optimal feedrate is rational in the curve parameter. For speed-dependentacceleration bounds, the optimal feedrate admits a closed-form expression in terms of a novel transcendental function whose values maybe efficiently computed, for use in real-time control, by a special algorithm. The optimal feedrate admits a real-time interpolatoralgorithm, that can drive the machine directly from the analytic path description. Experimental results from an implementation of thetime-optimal feedrate on a 3-axis CNC mill driven by an open-architecture software controller are presented. The algorithm is asignificant improvement over that proposed in Timar SD, Farouki RT, Smith TS, Boyadjieff CL. Algorithms for time-optimal controlof CNC machines along curved tool paths. Robotics Comput Integrated Manufacturing 2005;21:3753, since the addition of motorvoltage constraints precludes the possibility of arbitrarily high speeds along linear or near-linear path segments.r 2006 Elsevier Ltd. All rights reserved.Keywords: 3-Axis machining; Feedrate functions; Acceleration constraints; Time-optimal path traversal; Bang-bang control; Real-time interpolators1. IntroductionPrevious studies of time-optimal control in the fields ofrobotics 17 and CNC machining 810 were concernedwith the minimum-time traversal of a prescribed path by asystem with known dynamics and specified bounds on themotive-force capacity of its actuators. The solutions tosuchproblemscharacteristicallyincurabang-bangcontrol strategy, in which the output of at least onesystem actuator is saturated at each instant throughout thepath traversal. These studies typically assume actuatorswith constant and symmetric force limits (independent ofthe speed and direction of actuation), and generally do notaddress the question of the range of speeds over which theactuators can exert their maximum force.Fixed-field DC motors are common to most positioningand contouring applications in robotics and CNC machin-ing 11. Since their torque output is directly proportionalto the armature current, the constant symmetric torquelimits reflect the maximum current capacity of the motorarmature windings. Constant torque output is maintainedby continuously varying the armature voltage in relation tothe back EMF (proportional to the motor speed) orotherwise controlling the armature current supply 10.In addition to the armature current limits, the appliedarmature voltage may be subject to limits arising from themotor characteristics or armature power supply. Suchvoltage limits confine the ability of the motor to producethe maximum output torque to a finite range of speeds.Beyond this range, maximum applied armature voltagenot armature currentis the factor limiting the motorARTICLE IN PRESS/locate/rcim0736-5845/$-see front matter r 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.rcim.2006.07.002?Corresponding author.E-mail addresses: sdtimar (S.D. Timar),farouki (R.T. Farouki).torque output, resulting in a speed-dependent maximumtorque that decreases linearly with increasing motorspeed 10.In 3-axis machining, the maximum current capacity ofan axis drive motor imposes a constant acceleration limit atlower axis speeds, and the maximum voltage capacityimposes a speed-dependent acceleration limit at higher axisspeeds. The transition from current-limited to voltage-limited operation of the motor occurs at the axis transitionspeed. At speeds below the transition speed, the maximumaxis acceleration remains constant. At speeds greater thanthe transition speed, the maximum axis accelerationdecreases linearly with the axis speed, dropping to zero atthe axis no-load speed.To guarantee that time-optimal path traversals conformto both actuator current and voltage limits, algorithmsmust account for the regimes of both constant and speed-dependent acceleration limits on each machine axis. Thispaper generalizes the results of a previous study 9employing only constant acceleration bounds (an assump-tion that incurs arbitrarily high speeds if the path containsextended linear segments), and introduces new algorithmsto compute realistic time-optimal feedrates for CartesianCNC machines with axis drive motors subject to bothcurrentandvoltagelimits.Theinclusionofspeed-dependent acceleration bounds incurs significant, qualita-tive changes to many aspects of the earlier algorithm in9includingthesetoffeasiblefeedrateandfeedacceleration combinations v;a; the nature of the velocitylimit curve (VLC); the different types of possible switchingpoints; and the form of the feedrate function for extremalphase-plane trajectories. Nevertheless, for Cartesian CNCmachines with independently driven axes, it is still possibleto obtain an essentially closed-form solution for the time-optimal feedrate, given the ability to compute the roots ofcertain polynomial equations.We begin by reviewing DC motor operation in Section 2and the axis acceleration bounds in Section 3. Weintroduce the problem of minimum-time traversal ofcurved paths with constant and speed-dependent axisacceleration limits in Section 4, and we derive feedrateexpressions for constant and speed-dependent extremalacceleration trajectories. Feed acceleration limits, the VLC,and feedrate break points are then addressed in Sections57, respectively. Following a discussion of the feedratecomputation in Section 8, and the real-time CNC inter-polator algorithm in Section 9, we present details offeedrate computation and machine implementation resultsfor several examples in Section 10. Finally, Section 11summarizes our results and makes some concludingremarks.2. DC motor torque limitsAs background for understanding the nature of the axisacceleration bounds appropriate to Cartesian CNC ma-chines, we begin with a brief overview of the fixed-field DCmotors that are commonly used to drive small-to-mediummilling machines (see 10 for more complete details of theiroperation). The equations governing the operation of fixed-field motors areT KTI;E KEo;V E IR,i.e., the motor output torque T is proportional to thearmature current I, the back EMF E is proportional tothe motor angular speed o, and the applied armaturevoltage V is equal to the sum of the back EMF and thevoltage drop across the armature resistance R. Theproportionality factors KTand KE, called the torqueconstant and back EMF constant, are intrinsic physicalproperties of a given motor. From these expressions, onecan easily derive the motor torquespeed relationT Ts1 ?oo0?,(1)where Ts KTV=R is the stall torque, and o0 V=KEisthe no-load speed. Hence, the motor torque decreaseslinearly with increasing motor speed, from T Tsat o 0 to T 0 at o o0. See 12 for more complete details.At motor start-up and low speeds, the back EMF E issmall compared to the applied voltage V, and a current-limiting device is used to constrain the current I to an(approximately) constant maximum value Ilimto preventdamage to the armature windings. Hence, the motor torqueoutput remains constant at Tlim KTIlimthroughout thelow-speed range of operation.As the motor speeds up, the applied armature voltageeventually reaches the maximum motor or power supplyvoltage rating, Vlim. This occurs at the transition speed,defined byotVlim? IlimRKE.(2)For speeds greater than ot, the armature voltage (ratherthan the current) is the limiting factor on the motor torqueoutput. At the voltage limit, the torque T decreases linearlywith increasing motor speed o, dropping to zero when theno-load speed o0is attained.Fig. 1 depicts the motor constraints imposed by thecurrent and voltage limits, Ilimand Vlim, in the o;T planeforbothpositiveandnegativemotorspeeds.Theconstraints define two parallel strips, whose intersectionforms a paralellogram that defines the feasible regime ofDC motor operation. All admissible combinations ofmotortorqueandspeed,consistentwiththegivenarmaturecurrentandvoltagelimits,liewithinthisparalellogram.The portions of the paralellogram extending beyond theno-load speed in each direction (oo ? o0and o4 o0)correspond to regenerative braking of the motor, whichimplies application of an external torque. Since no suchtorque is available in the context of CNC machine drivemotors, the range of feasible torque/speed states is reducedARTICLE IN PRESSS.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579564to indicate the no-load speed as the maximum motor speed,yielding the six-sided parallelogram shown in Fig. 1.The six-sided parallelogram defines three distinct DCmotor speed ranges, each with distinct minimum andmaximum torque limits, namely:?Tlimo0 oo0? otpTp Tlimfor ? o0pop ? ot,?TlimpTp Tlimfor ? otpop ot,?TlimpTp Tlimo0? oo0? otfor otpop o0.3. Axis acceleration limitsIn high-speed machining 8,13,14 inertial forces maydominate cutting forces, friction, etc., especially for toolpaths of high curvature. Accounting for the axis inertias,the axis speeds and accelerations are proportional to themotor speeds and motor torques, respectively. Consider,say, the x-axis. If it has effective mass Mxand is actuatedby a drive motor through a ball screw of modulus Kx(i.e.,the linear axis velocity vxis related to the motor angularspeed o by vx o=Kx), the axis acceleration correspond-ing to motor torque T is ax KxT=Mx. Noting that thefeedrate may be regarded as a vector of magnitude v anddirection given by the unit path tangent t tx;ty;tz, wehave vx txv and the motor rotational speed is o Kxtxv.Hence, the torque limits derived above are equivalent tothe x-axis acceleration limits? Axv0 vxv0? vtpaxp Axfor ? v0pvxp ? vt,? Axpaxp Axfor ? vtpvxp vt,? Axpaxp Axv0? vxv0? vtfor vtpvxp v0,3where vtis the axis transition speed, v0is the axis no-loadspeed, and we define Ax KxTlim=Mx. By virtue of thespeed-dependent acceleration limits, the axis speed vxalways remains in the interval ?v0;v0?.Within the axis speed range vx2 ?vt;vt, the mini-mum and maximum axis acceleration limits are both fixed,and hence this is referred to as the constant limits regime forthe x-axis. The axis speed ranges vx2 ?v0;?vt andvx2 vt;v0, for which one acceleration limit is fixedand the other is speed dependent, are called the mixedlimits regimes for the x-axis.In the constant limits regime, the acceleration boundsmay be written as axAx, with ax ?1. For the mixed limitsregime, the acceleration bounds may be expressed in theformAxgxv0? vxv0? vtand? gxAx,where gx ?1 for vx2 ?v0;?vti.e., txo0, and gx 1forvx2 vt;v0i.e.,tx40.Similarconsiderationsapply to the y- and z-axis.During a path traversal, each axis operates within one ofits acceleration limit regimes independently of the otheraxis, and each may switch between the acceleration limitregimes in accordance with variations in the tool pathgeometry and feedrate. Consequently, there are fourpossiblecombinationsofacceleration-limitedregimesamong the x-, y-, z-axis (see Table 1). For a planar curve,ARTICLE IN PRESSTTFig. 1. Left: the maximum current and voltage limits impose constant and speed-dependent torque limits, respectively, forming a four-sided parallelogram(shaded) of feasible motor torque/speed values. Right: since the motors that drive CNC machine axes will not exceed the no-load motor speed, the regionof feasible torque/speed values is truncated to form a six-sided parallelogram.Table 1The four possible combinations of acceleration-limited regimes for a 3-axisCNC machine (here a;b;c denotes any permutation of the axes x;y;z)AxisabcconstantconstantconstantmixedconstantconstantmixedmixedconstantmixedmixedmixedS.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579565involving motion of only two machine axes, there are threepossible combinations: constant/constant, constant/mixed,and mixed/mixed. Each combination of acceleration limitsincurs a specific analysis to compute the time-optimalfeedrate. The case in which all axes are in the constantregime is covered by our earlier study 9, but casesinvolving one or more of the axes in the mixed regimehave not been previously addressed.4. Time-optimal feedratesConsider a path described by a degree-n Be zier curverx Xnk0pknk?1 ? xn?kxk;x 2 0;1?(4)with control points pk xk;yk;zk, k 0;.;n 15. If sdenotes arc length measured along the curve, we define theparametric speed bysx jr0xj dsdx.The unit tangent and (principal) normal vectors and thecurvature of (4) are defined byt r0s;n r0? r00jr0? r00j? t;k jr0? r00js3(5)and, conversely, with s0 r0? r00=s we may writer0 st;r00 s0t s2kn.(6)Now suppose we traverse the curve with feedrate (speed)specified by the function vx. Since derivatives with respectto time t and the parameter xwhich we denote by dotsand primes, respectivelyare related byddtdsdtdxdsddxvsddx,the velocity and acceleration vectors at each point are givenbyv _ r vt;a r _ vt kv2n.(7)The tangential component _ vt of a vanishes if v constant,while the normal (centripetal) component kv2n vanishes ifk 0. The time derivative of the feedrate (the feedacceleration) is given in terms of x as _ v vv0=s.We wish to minimize the traversal time along rx,starting and ending at rest, subject to acceleration limits ofthe form (3) and analogous expressions for the othermachine axes. These requirements can be phrased in termsof the following optimization problem:minvxT Z10svdx(8)such thatAi;minpaixpAi;maxfor x 2 0;1?,where i x;y;z refers to each of the Cartesian componentsax;ay;azof a. As noted in Section 3, the axis accelerationbounds Ai;min, Ai;maxare of the form?Ai;AiorAigiv0? viv0? vt;?giAi.4.1. Constant acceleration trajectoriesFrom the relations (5), (7), ss0 r0? r00, and _ v vv0=s,we may writea vv0s2r0v2s3sr00? s0r0.Foragivencurverx xx;yx;zxthex-axiscomponent (say) of the acceleration a is defined byaxq02s2x0qs3sx00? s0x0,(9)where we write q v2, since it is convenient to work withthe square of the feedrate (see 9 for further details).During an extremal acceleration phase under constantacceleration limits, one component of the acceleration isequal to plus or minus the corresponding bound, acondition that yields a linear differential equation for q.If x is the extremally accelerating axis, this equation admitsa closed-form solution for the (squared) feedrate, namelyq sx0? ?2C 2axAxx,(10)wheretheintegrationconstantCisdeterminedbyspecifying a known point x?;qx? on the trajectory:C x0x?=sx?2qx? ? 2axAxxx?. Further details ofthe solution method for (10) may be found in 9.4.2. Speed-dependent acceleration trajectoriesConsider the determination of the feedrate v when the x-axis (say) executes an extremal acceleration defined by aspeed-dependent acceleration bound, of the form describedabove. The differential equation governing the feedrateunder such circumstances istx_ v knxv2AxZv0txv ?gxAxZ 0,(11)where we introduce the constantZ 1 ?vtv0.Eq. (11) is a first-order, non-linear differential equationwith variable coefficients. It may be written exclusively interms of x asvv0x00x0?s0s?v2AxZv0sv ?gxAxZs2x0 0.ARTICLE IN PRESSS.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579566To obtain a closed-form integration of this equation, wenote thatvv0x00x0?s0s?v212sx0? ?2ddxx0sv?2.Hence, since g2x 1, we obtainddxx0svv0?2 2gxAxZv20x01 ? gxx0svv0?.Writing u x0=sv=v0, this givesududxgxAxZv20x01 ? gxu,which is amenable to separation of variables, givingZudu1 ? gxugxAxZv20Zx0dx.Noting again that g2x 1, this can be integrated to obtain1 ? gxu ? ln1 ? gxu gxAxZv20x c,the integration constant c being determined from a knowninitial condition. We note that gxu gxx0=sv=v0 satisfies0pgxup1, since 0pv=v0p1, ?1px0=sp 1, and gxhasthe same sign as x0=s. Hence, the argument of thelogarithm occurring above is between 0 and 1.Now let ck be the transcendental function that isdefined implicitly as the solution of the equationck ? lnck k.(12)By differentiating, we see thatdcdk ?ck1 ? ck,and hence the function ck is monotone decreasing if itsrange is confined to 0pckp1. The correspondingdomain is 1pkp1. Using the function c, we can writethe feedrate explicitly in terms of the curve parameter x asvx gxv0sxx0x1 ? cgxAxZv20xx c?.We regard ck as a basic transcendental function, ofsimilar stature to the trigonometric or logarithmic func-tions. To use it in the context of real-time motion control,an efficient means to evaluate this function is required.Re-writing (12) in the formck exp?k expck(13)yields the iteration sequence for ck defined bycr exp?k expcr?1;r 1;2;.(14)with a suitable starting approximation c0. For 1oko1and 0ocko1, the derivative of the right-hand side of(13) with respect to c is of magnitude less than 1, and hencethe iteration (14) is convergent for all values of k andstartingapproximationsc0withintheserangesseeTheorem 6.5.1 of 16. The convergence can be accele-rated, after the first step, by use of the Aitken extra-polation 16:cr cr?cr? cr?12cr? 2cr?1 cr?2.(15)To estimate starting values c0, we use linear interpolationbetween a sequence of pre-computed values (see Table 2).5. Set of possible v;a valuesIn the computation of time-optimal feedrates, thecharacterization of the set of possible combinations v;aof feedrate and feed acceleration at each point along thepath, as determined by the prescribed physical constraintson each of the machine axes, plays an important role. Fromthis we may determine the VLC vlimxi.e., the maximumfeedrate consistent with the axis constraints, and the rangeaminx;vpapamaxx;v of possible feed accelerations ateach feedrate v less than vlimx.In the case of constant acceleration bounds on all axes,the acceleration constraints at each curve point x describestrips in the v2;a plane, bounded by parallel line pairs.The intersection of these strips defines a parallelogram,whose interior constitutes the set of feasible v2;a values,and whose right-most vertex defines vlimx. For eachfeedrate v less than vlimx, the upper parallelogramboundarydefinesthemaximumfeedaccelerationamaxx;v, and the lower parallelogram boundary definesthe minimum feed acceleration aminx;v. We refer thereader to 9 for complete details.In the case of mixed acceleration bounds, either thelower or the upper constraint involves both v and v2, aswell as a, and is thus not describable by a linear relation inthe v2;a plane. Therefore, we choose to characterize theset of admissible states in the v;a plane, rather than theARTICLE IN PRESSTable 2Nodal values of the transcendental function defined by Eq. (12)kck1.001.0000001.020.8131051.050.7161891.100.6168171.200.4932391.500.3017102.000.1585945.000.006784A piecewise-linear approximation based on these values is employed togenerate starting approximations for the iteration defined by (14) and (15).S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579567v2;a plane. At curve points x corresponding to mixedacceleration bounds, the constraints for each axis defineregions bounded by pairs of parabolas in the v;a plane.The intersection of these parabolic strips for each axisdefines a star-shaped region, whose interior constitutes theset of admissible v;a combinations. The right-most pointof this region defines vlimx, while for each vovlimxitsupperandlowerboundariesdefinetherangeaminx;vpapamaxx;v of feasible feed accelerations.We derive below the constant and mixed axis acce-leration bounds as feed acceleration limits written in theformax;v a0x a1xv a2xv2,where a1x ? 0 in the case of constant accelerationbounds. By analyzing the a0x and a2x coefficients foreach bound, the set of feasible v;a states can becompletely characterized at any curve point x. Note thatthe feedrate v is, by assumption, a positive quantityi.e.,the curve is always traversed in the sense of increasing xand hence only the portion of the set of feasible v;a valuesin the right half of the plane vX0 is of interest.For simplicity, we consider henceforth only motion inthe x;y plane with equal acceleration bounds Ax AyA; say and identical transition and no-load speeds, v0andvt, on both axes. The tangent and normal are then t x0;y0=s and n ?y0;x0=s, and we write the (signed)curvature ask hs3with h x0y00? x00y0.(16)5.1. Constant axis acceleration boundsWith ax;ay ?1, the feed acceleration bounds imposedby the x-, y-axis area axAsx0 ky0x0v2;a ayAsy0? kx0y0v2.(17)In the v;a plane, each pair of acceleration constraintsdefines two parabolas that are symmetric about the a-axisand do not intersect, are both convex or concave upward,and have a0values of equal magnitude but opposite sign.This description of the set of states compatible withconstant acceleration bounds is equivalent to that givenpreviously in 9, except that the set is specified here in thev;a plane, rather than in the v2;a plane.For the acceleration constraint pair associated with eachmachine axis, the constraint with a040 defines themaximum feed acceleration amax, and that with a0o0defines the minimum feed acceleration amin. Sign changes inthe a0expressions for an axis constraint pair arise at pathturning points corresponding to that axis 9. Between suchpoints, the signs of x0and y0identify the constraints witha040 and a0o0 among the pair.5.2. Mixed axis acceleration boundsIn the mixed regime, the feed acceleration boundsimposed by the x-axis at its constant and velocity-dependent acceleration limits area ?gxAsx0 ky0x0v2;a gxAZsx0?AZv0v ky0x0v2(18)with gx ?1, and likewise for the y-axis acceleration wehavea ?gyAsy0? kx0y0v2;a gyAZsy0?AZv0v ? kx0y0v2(19)with gy ?1. For fixed gx, gyEqs. (18) and (19) each definea pair of parabolas in the v;a plane with the sameconvexity but unequal a0values of opposite sign. More-over, the presence of the v term in the latter constraints in(18) and (19) incurs asymmetry about the a-axis in eachparabola, and asymmetry between the two parabolasdefined by each axis constraint pair.Since they have identical v2terms, the two x-axisconstraint parabolas intersect in a single point in thev;a plane, with coordinatesv gx2v0? vtsx0;a ks2y0x032v0? vt2? gxAsx0.Likewise, the y-axis constraints intersect at the point withcoordinatesv gy2v0? vtsy0;a ?ks2x0y032v0? vt2? gyAsy0.Since gx, gyhave the same sign as x0, y0, respectively, thesepoints lie in the right half of the v;a plane. Hence, theconstraints for a single axis in the mixed limits regimedefine a set of feasible v;a values that is closed in the righthalf of the v;a plane, and remains open in the left half ofthe plane.Characteristic of the mixed constraints, the constraintwith the additional v term always has a040, and theconstraint without this term always has a0o0. This isconsistent with the axis acceleration limits discussed inSection 3 where, within the speed domains ?v0;?vt andvt;v0, the velocity-dependent acceleration limits therate of axis speed increase, and the constant accelerationlimits the rate of speed decrease, in either direction of axistravel.5.3. Minimum and maximum feed accelerationAt each curve point x, the pairs of parabola constraintsin the v;a plane incurred by the x- and y-axis always haveopposite convexities, as indicated by the signs of their a2coefficients. Also, the intercepts a0of the two parabolasdefined by a single axis always have opposite signs. Hence,ARTICLE IN PRESSS.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579568the two pairs of parabolas will always yield a closed finiteset of admissible v;a values, with well-defined extrema inv and a.For both the constant/mixed and mixed/mixed con-straint combinations, the upper bound amaxx;v on thefeed acceleration is defined by either one or both of the x-,y-axis constraints with a040, and the lower boundaminx;v is defined by either one or both of the x-, y-axisconstraints with a0o0.Labelling the two constraints that have a040 withsuperscripts (1) and (2), constraint (1) alone defines amaxwhen a10oa20, a12o0 and a2240. However, constraints(1) and (2) both define amaxover appropriate domains in vwhen a1240 and a22o0. If instead a104a20, the conversea12, a22relationships determine amax, i.e., constraint (1)defines amaxwhen a1240, a22o0 and constraints (1) and(2) both define amaxwhen a12o0, a2240.Likewise, labelling with superscripts (3) and (4) the twoconstraints that have a0o0, constraint (3) alone definesaminwhen a304a40, a3240 and a42o0. However, con-straints (3) and (4) both define aminover appropriatedomains in v when a32o0 and a4240. If instead a30oa40,the converse a32, a42relationships determine amin.Figs. 2 and 3 illustrate the possible configurations ofthe parabolas that define the lower and upper feedacceleration limits, aminand amax, in the case of constant/mixed and mixed/mixed acceleration bounds (equi-orienta-tion points and pseudo equi-orientation points are definedin Section 7).ARTICLE IN PRESSvavavaFig. 2. For the case of constant/mixed acceleration bounds, the x- and y-axis minimum and maximum feed accelerations aminand amaxare indicated atthree types of points on the curve rx: a generic point (left); an equi-orientation point (center); and a pseudo equi-orientation point (right).vavavaFig. 3. For the case of mixed/mixed acceleration bounds, the x- and y-axis minimum and maximum feed accelerations aminand amaxare indicated at threetypes of points on the curve rx: a generic point (left); an equi-orientation point (center); and a pseudo equi-orientation point (right).vavlimFig. 4. The maximum feedrate vlimconsistent with all the axis accelerationconstraints is identified by the right-most point of the set of feasible v;avalues at each x. A unique feed acceleration aVLCis defined by this point.S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 5635795696. Velocity limit curveAt each curve point rx, the maximum feedrate vlimxconsistent with the axis acceleration constraints is specifiedby the right-most point of the set of feasible v;a values(see Fig. 4). The variation of this maximum feedrate vlimwith x is called the VLC in the x;v phase plane. Acharacteristic feature of states x;vlimx on the VLC is thatthe minimum and maximum feed accelerations are equal:amin amax. The unique value of the feed acceleration ateach point of the VLC is denoted by aVLC.The function vlimx is piecewise analytic, the expressionsthat define each of its segments being dependent upon theprevailing configuration of the axis acceleration constraintsin the v;a plane over appropriate intervals in x. The signof the quantity kx0y0is an important parameter incharacterizing these segments. We analyze in detail beloweach of the possible cases.6.1. Analytic expressions for VLC segmentsIf constant acceleration limits hold on both the x- and y-axis, vlimis defined by equating expressions (17) to obtainvlimffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAs2hayx0? axy0r,(20)choosing ax ?signx0, ay signy0 when kx0y040, andax signx0, ay ?signy0 when kx0y0o0.If constant acceleration limits hold on the x-axis, andmixed acceleration limits hold on the y-axis, vlimis given byvlimAsx0y02Zv0h?1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4Zv20hAx0y02gyx0? Zaxy0s#(21)with ax ?signx0 when kx0y040. Otherwise,vlimffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi?As2hgyx0 axy0r(22)with ax signx0 when kx0y0o0.If mixed acceleration limits hold on the x-axis, andconstant acceleration limits hold on the y-axis, vlimis givenbyvlimffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAs2hayx0 gxy0r(23)with ay signy0 when kx0y040. Otherwise,vlimAsx0y02Zv0h1 ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4Zv20hAx0y02Zayx0? gxy0s#(24)with ay ?signy0 when kx0y0o0.Finally, if mixed acceleration limits hold on both the x-and y-axis, thenvlimAsx0y02Zv0h?1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4Zv20hAx0y02gyx0 Zgxy0s#if kx0y04025andvlimAsx0y02Zv0h1 ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ?4Zv20hAx0y02Zgyx0 gxy0s#if kx0y0o0.26Expressions (20)(25) specifying the VLC vlimx undervarious combinations of constant and mixed accelerationlimits on the x-, y-axis, and signs of kx0y0, are all of theformvlimffiffiffiffiPporvlim Qd1 d2ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp?,whered1;d2 ?1andP;Q;Rareknownrationalfunctions of x in each case. The derivatives of theseexpressions arev0limP02ffiffiffiffiPpandv0lim d1Q0 d22Q01 R QR02ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp,where P0, Q0, R0are easily obtained from the known formsof P, Q, R.6.2. Switching points on the VLCThe identification of switching points on the VLC, wherethe time-optimal feedrate may change between integrationof aminand amaxtrajectories, is a key pre-processing step9. Switching points on the VLC are either critical points ortangency points. As with the constant/constant accelerationlimits case 9, critical points are tangent discontinuities ofthe VLC, which also arise at path inflections and turningpoints in both the constant/mixed and mixed/mixedconstraint combinations. A critical point is a viableswitching point only if the feed acceleration limits aminand amaxat that point do not exceed the VLC slopes on theleft and right of the critical point, respectively.Tangency points correspond to transitions between VLCsink and source segments. They may be computed as rootsof the equationvlimv0lim? saVLC 0(27)on each VLC segment. The constant/constant accelerationbounds case was treated in 9. For the constant/mixed andmixed/mixed constraints cases, aVLCcan be determined bysubstituting the expression for vliminto either of theappropriate axis acceleration constraints (both constraintsyield the same aVLCexpression), after which the roots of(27) can be computed.When constant acceleration bounds hold on both the x-and y-axis, the unique feed acceleration at the VLC isfound by substituting (20) into either of Eqs. (17) to obtainaVLC Aaxx0 ayy0s.When constant acceleration bounds hold on the x-axis andmixed acceleration bounds hold on the y-axis, the uniquefeedaccelerationaVLCexpressionsforkx0y040andARTICLE IN PRESSS.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579570kx0y0o0 are, respectively,aVLCaxAs ky0Q2d1 d2ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp2x0andaVLCaxAs ky0Px0.When mixed acceleration limits hold on the x-axis andconstant acceleration limits hold on the y-axis we have, forkx0y040 and kx0y0o0, respectively,aVLCayAs ? kx0Py0andaVLCayAs ? kx0Q2d1 d2ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp2y0.Lastly, when the x- and y-axis are both under mixedacceleration limits, the feed acceleration aVLCfor kx0y040and kx0y0o0 is, respectively,aVLC?gxAs ky0Q2d1 d2ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp2x0andaVLC?gyAs ? kx0Q2d1 d2ffiffiffiffiffiffiffiffiffiffiffiffi1 Rp2y0.In the above expressions, ax;ay, gx;gy, d1;d2and P;Q;Rmust be chosen in accordance with the appropriateexpression for vlimin Section 6.1.7. Feedrate break pointsBreak points of the optimal feedrate occur where thefeed acceleration limits aminx;v and amaxx;v are non-differentiable with respect to x. Such points arise when: (1)a10 a20; (2) a30 a40; and (3) a12 0 or a22 0.7.1. Equi-orientation pointsThe first type of break point corresponds to path equi-orientation points 9 for both the constant/mixed andmixed/mixed constraint combinations. The second type ofbreak point corresponds to equi-orientation points for themixed/mixed constraints. For constant/mixed constraints,however, this point corresponds to pseudo equi-orientationpoints, defined the roots ofy02? Z2x02 0andx02? Z2y02 0(28)for x 2 0;1?. Equi-orientation points and pseudo equi-orientation points are generalizations of the conditionsa10 a20and/or a30 a40for all possible accelerationconstraint combinations.7.2. Inflection pointsThe third type of break point corresponds to pathinflections, where k 0, or turning points where x0 0 ory0 0. At inflection points, the centripetal accelerationcomponent vanishes, and hence the constant accelerationbounds becomea axAsx0anda ayAsy0,while the mixed acceleration bounds on the x- and y-axisbecomea ?gxAsx0anda AZgxsx0?vv0?,a ?gyAsy0anda AZgysy0?vv0?.At turning points, the feed acceleration component isabsent from one axis constraint pair and the centripetalcomponent is absent from the other pair. At turning pointswith tangent t 0;?1 and ?1;0, the constant x- and y-axis acceleration bounds becomev ffiffiffiffiffiffiffiffiffiffiffiffiffi?axAkr;a ?ayAandv ffiffiffiffiffiffiffiffiffiffiffiffiffi?ayAkr;a ?axA,respectively. At turning points with tangent t 0;?1 themixed x- and y-axis acceleration bounds becomev ffiffiffiffiffiffiffiffiffiffi?AZks;ffiffiffiffiffiffiffiffi?Akranda AZ1 ?vv0?;?A,while at turning points with t ?1;0, they becomev ffiffiffiffiffiffiffiffiffiffi?AZks;ffiffiffiffiffiffiffiffi?Akranda AZ1 ?vv0?;?A.The feed acceleration bounds aminx;v, amaxx;v are C0functions of x, with break points arising at path equi-orientation points, pseudo equi-orientation points, inflec-tions, and turning points. Between such points, theparticular configuration of the x-, y-axis accelerationconstraints in the v;a plane determines the exact formof the feed acceleration bounds.7.3. Transition pointsWhen the lower or upper boundary (aminor amax) of theset of possible v;a values is defined by two constraints, achange in the time-optimal feedrate expression will occurwhen the current state x;v coincides with a point ofintersection of the two constraints. Such points are calledtransition points19, and may occur under any of the threepossible axis acceleration bound combinations: constant/constant, constant/mixed, and mixed/mixed.Unlike the break points described above, which can beidentified in terms of the curve geometry alone, theoccurrence of transition points depends on the currentstate x;v when integrating aminand amaxtrajectoriesto compute the time-optimal feedrate function. TheyARTICLE IN PRESS1Transition points should not be confused with points where an axisshifts between the constant and mixed acceleration bounds, at the axistransition speed vt.S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579571correspond to a coincidence of the current state x;v with apoint defined by the intersection of two constraints definingthe lower or upper boundary, aminx;v or amaxx;v, of theset of feasible v;a states at any curve point x. We use thelabelling of constraints in the v;a plane introduced inSection 5.3i.e., the two constraints that have a040 arelabelled with superscripts (1), (2) and the two constraintsthat have a0o0 are labelled with superscripts (3), (4).When the x-axis is subject to constant acceleration limitsand the y-axis is subject to mixed acceleration limits, thevalue of v in the right half of the v;a plane at which thetwo upper constraints intersect isv1;2Asx0y02Zv0h?1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4Zv20hAx0y02gyx0? Zaxy0s#,where ax signx0. Similarly, the value of v in the righthalf of the v;a plane at which the two lower constraintsintersect isv3;4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi?As2hgyx0 axy0r,where ax ?signx0.When the y-axis is subject to constant limits and the x-axis is subject to mixed limits, the positive value of v atwhich the two upper constraints intersect isv1;2Asx0y02Zv0h1 ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4Zv20hAx0y02Zayx0? gxy0s#where ay signy0. Likewise, the positive value of v atwhich the two lower constraints intersect isv3;4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAs2hayx0 gxy0r,where ay ?signy0 (Fig. 5).Finally, when the x- and y-axis both subject to mixedacceleration limits, the positive values of v at which thetwoupperandtwolowerconstraintsintersectare,respectively,v1;2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAs2Zhgyx0? gxy0sandv3;4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAs2hgxy0? gyx0r.Transition points occur only under certain path andactuator conditions, as characterized by the constraint a0and a2coefficients. The upper boundary amaxexhibits anintersection of two constraints whena104a20;a12o0;a2240or the converse conditions. Similarly, the lower boundaryaminexhibits an intersection whena304a40;a32o0;a4240or the converse conditions. The two sets of conditions forthe upper and lower boundaries of the set of feasible v;avalues differ in the order of the intersecting accelerationconstraints before and after the intersection point.8. Feedrate computationEach combination (i.e., constant/constant, constant/mixed, or mixed/mixed) of the x- and y-axis accelerationconstraintsyieldsspecificfeedaccelerationboundsaminx;v, amaxx;v and a feedrate limit vlimx, consistentwith that combination. Each segment of the optimalfeedrate vx between breakpoints corresponds to a specificcombination of acceleration constraints, with shifts be-tween combinations occurring at the axis transition speeds.8.1. Time-optimal feedrate algorithmIn accordance with the bang-bang principle of time-optimalcontrol,theoptimalfeedrateisdefinedbyalternatelyintegratingtrajectoriesoftheform_ v aminx;v and _ v amaxx;v in the x;v phase plane. TheVLC divides this plane into feasible and infeasible systemstates, below and above the VLC. Changes in the analyticform of the optimal feedrate vx occur at switching points(alternations between aminand amaxtrajectories, on orARTICLE IN PRESSvav(1,2)vav(3,4)Fig. 5. For a set of feasible v;a combinations at a given curve point x, the two feedrates v1;2and v3;4that identify transition points on the upper (amax)and lower (amin) boundaries, respectively, are indicated.S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579572below the VLC) and break points of the type identified inSection 7.For details of the time-optimal feedrate algorithm, werefer the reader to our previous study 9 of the constant/constant bounds case. This algorithm can be readilygeneralized to accommodate the new types of switchingpoints and extremal acceleration trajectories, correspond-ing to constant/mixed and mixed/mixed constraint combi-nations.8.2. Transition and no-load feedratesIn the present context, a new type of feedrate break pointmust be taken into consideration, associated with a changeinthenatureconstant/constant,constant/mixed,ormixed/mixedof the prevailing acceleration bounds at aparticular point x;v. Such a point becomes the initial statefor feedrate integration under the ensuing combination ofacceleration bounds. We call this new type of feedratebreak point a constraint change point.Changes between constant/constant, constant/mixed,and mixed/mixed acceleration combinations occur at thex- and y-axis transition speeds,vt;x vtsjx0jandvt;y vtsjy0j.(29)A point x;vx such that vx vt;xor vx vt;yis a breakpoint of the optimal feedrate, and serves as the initialcondition for feedrate computation under the new combi-nation of axis acceleration bounds. Similarly, we define theno-load feedrate byv0;xy v0minsjx0j;sjy0j?.This represents the maximum feedrate attainable under thespecified current and voltage limits of the axis drivemotors. Note that vt;x, vt;y, v0;xydepend on position xalong the path.9. Real-time interpolatorAs a pre-processing step, the curve is subdivided intosegments identified by their unique active accelerationconstraints. Associated with each of these segment is afeedrate expression of the form (10) or (11), correspondingto constant or speed-dependent extremal axis acceleration.The function of the real-time CNC interpolator is tocompute, from the specified path and feedrate, referencepoints that indicate the commanded machine positionwithin each sampling interval Dt of the digital controller.The equation that determines the reference-point para-meter value xkat time kDt iskDt Zxk0svdx,(30)since dx=dt ds=dtdx=ds v=s. Segments that corre-spond to constant axis acceleration limits admit closed-form reduction of the above integral to obtain (with, say, xas the limiting axis)kDt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC 2axAxxkpA K,(31)with integration constant K such that t 0 when x 0note that each curve segment is re-parameterized to havedomain x 2 0;1?.Segments corresponding to speed-dependent accelera-tion limits do not admit closed-form reduction of theintegral (30). Instead, we employ 17 a truncated Taylorseries expansion of xt about t tkto estimate parametervalues at successive kDt interrupts:xk1 xkvsDt 12vv0s ? v2s0s3Dt2 ?,(32)starting with x0 0. In (32) it is understood that the Taylorseries coefficients are evaluated at xk.10. Experimental resultsComputation of the time-optimal feedrate function isperformed by an off-line program that employs numericalroot-solving methods, based upon the numerically stableBernstein polynomial representation 18,19 on x 2 0;1?.This program receives as input a parametric curve, axisacceleration bounds, and the axis transition and axis no-load speeds. It outputs a segmentation of the curve, thefollowing information being associated with each segment:the active constraint (indicated by the axis identity andthe sign and nature of the acceleration constraint), thefeedrate integration constant, and the starting and ending xvalues. These data suffice for the real-time interpolator torealize the time-optimal feedrate over the entire path.The complexity of the optimal feedrate computation,under both constant and mixed axis acceleration bounds, isillustrated by three examples. In order to highlight essentialdifferences from the simpler algorithm in 9, we employ aplanar parabolic tool path with just a single turning pointand no inflections or equi-orientation points, and amaximumaxisaccelerationmagnitudeofA 20;000in=min2and speed ratio vt=v012for all threemachine axesthe algorithm works equally well for spatialpaths and independent A and vt=v0values on each axis. Thethree examples below differ only in the axis transitionspeedswerefertothechoicesvt 250,130,and75in=min as the high, medium, and low cases,respectively. Following presentation and discussion of thecomputed feedrates, we demonstrate the results of theirimplementation on a 3-axis CNC milling machine operat-ing in conjunction with the real-time interpolator algo-rithms described in Section 9.10.1. Optimal feedrate resultsFor each of the examples, we present four plotscomprising: the tool path in the x;y plane (top left); theARTICLE IN PRESSS.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579573computed time-optimal feedrate (solid curve) in the x;vphase plane, together with the VLC (dashed) and the x-axis(squares) and y-axis (triangles) transition and no-loadfeedrates (top right); and the x- and y-axis accele-ration versus speed trajectories in the vx;ax and vy;ayplanes (bottom left and right). In each plot, the optimalfeedrate switching points, numbered successively, are alsoindicated.Turning our attention to Fig. 6 for the high limit casewith vt 250in=min and v0 500in=min, we see that theoptimal feedrate always remains below the x- and y-axistransition feedrates, and the axes thus remain in theconstant/constantacceleration boundsregime. Hence,acceleration limits on the axescoupled with the pathcurvatureare the dominant actuator constraints, and theanalysis reverts to the principles described in 9. Becausethe curve is slightly asymmetric, the x-axis exhibits unequalspeed and acceleration values vx;ax between x 0 and 1.In Fig. 7 we show results for the medium case withvt 130in=min and v0 260in=min. In this case, theinitial amaxand final aminintegration trajectories both crossthe y-axis transition feedrate, marking changes from theconstant/constanttoconstant/mixedaxisaccelerationbounds. Where the amaxintegration crosses, there is a shiftfrom constant to speed-dependent acceleration, whichreduces the achieved feedrate and delays the occurrenceof switching point x1. Where the amaxintegration crossesthe y transition feedrate, the constraints intersecting todefinevlimabruptlychangewhenshiftingfromtheconstant/constant to mixed/constant x;y acceleration limitcombination,incurringadiscontinuityintheVLC.Because the constraints that define vlimdo not changewhen changing between the mixed/constant to constant/constant combination along the aminintegration, the VLCis continuous where this trajectory crosses the y transitionfeedrate.ARTICLE IN PRESS-3-3-2-2-1-10123024-113501231x iny in01002003004005000112231v in/min-600-400-2000200400600012310410401vx in/minvy in/minax in/min2ay in/min2-600-400-2000200400600-3-2-1012301+1-3-32-3+1+1-3+Fig. 6. The path, optimal feedrate, and axis operating states for the high axis transition speed, with switching points x1, x2, x3(the ? superscriptsindicate states before and after a switching point). Throughout the traversal, each axis remains within the window delimited by ?vt;?A, yielding anoptimal feedrate composed solely of constant axis acceleration segments, and a continuous VLC.S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579574Finally, Fig. 8 presents results for the low case withvt 75in=min and v0 150in=min. The initial amaxandfinal aminintegration trajectories cross both the x- and y-axistransitionfeedrates,markingchangesfromtheconstant/constant to constant/mixed to mixed/mixed x-,y-axis acceleration bounds combinations. The VLC, lyingabove the y-axis no-load feedrate, is never intercepted byan amaxor aminintegration trajectory. Instead, the initialamaxintegration asymptotically approaches the y-axis no-load feedrate until it intersects the final amintrajectory atswitching point x1. Again, the VLC exhibits discontinuitieswhere changes in the constraints intersecting to define vlimoccur between the three acceleration limit combinations.As the examples indicate, the feedrate computationfollows the algorithm in 9 and the resulting feedratereflects the bang-bang control strategy characteristic ofminimum-timetraversalofsystemswithconstrainedmotive force, i.e., the acceleration magnitude on eitherthe x- or y-axis attains the prescribed bounds in theconstantorspeed-dependentregimesthroughoutthemotion. Note that, if vt=v0! 1, the time-optimal feedrateresembles that computed under the assumption of constantaxis acceleration and velocity constraints 10, and theconvention for designating feedrate switching points in 10agrees with feedrate switching point locations arisingnaturally under constant and speed-dependent accelerationbounds.10.2. Optimal feedrate implementationWe present implementation results for the high, medium,and low feedrate cases illustrated in Figs. 68. Theexperiments were performed on a precision 3-axis CNCmill, driven by an open-architecture software controllerthat permits incorporation of the real-time interpolatoralgorithms described in Section 9. The digital controller hasARTICLE IN PRESS-3-3-2-2-1-10123024-113501x iny in01002003004005000112321231v in/min-400-2000200400012310410401vx in/minvy in/minax in/min2ay in/min2-400-2000200400-3-2-1012301+1-3-32-3+1+1-3+Fig. 7. The path, optimal feedrate, and x-, y-axis operating states for the medium axis transition speed, with switching points x1, x2, x3indicated.During the traversal, the y-axis shifts between acceleration bound regimes, yielding an optimal feedrate composed of both constant and speed-dependentacceleration segments and a discontinuous VLC.S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579575a sampling frequency of 1024Hz. By computing first andsecond differences of the real-time position data outputfrom the interpolator during each run, the velocity,feedrate, and acceleration performance can be assessed aposteriori.For the chosen test curve, Figs. 10, 12, and 14 show thevelocity and acceleration data from runs at the high,medium,andlowtransitionspeeds,withA 20;000in=min2and vt=v012. We show separately the xand y components of the velocity for each run, obtained bydifferencing of measured position data. Figs. 9, 11, 13 showthe feedrate magnitudes for the specified test curve. Notethat, in these plots, the independent variable is the elapsedtime t, rather than the curve parameter x (Figs. 914).Figs. 10, 12, and 14 illustrate the x and y accelerationcomponents for each case, obtained by second differencingof measured position datathe noise is a consequenceof the differencing operation. It is apparent that theoptimal feedrate derived by the algorithm of Section 4 ingeneral corresponds to a bang-bang control strategyi.e., the acceleration magnitude on either the x- or they-axis attains the prescribed constant or speed-dependentbound. In Figs. 12 and 14 we see that the velocity ofthe axis under a speed-dependent acceleration boundARTICLE IN PRESS-3-3-2-2-1-10123024-113501x iny in010020030040050001111v in/min-200-1000100200012310410401vx in/minvy in/minax in/min2ay in/min2-200-1000100200-3-2-1012301+1-1+1-Fig. 8. The path, optimal feedrate, and axis operating states for the low axis transition speed, with a single switching point x1. During the traversal, bothaxes shift between acceleration bound regimes. The VLC in this case exhibits several discontinuities.0.00.51.01.52.02.5050100150200250time secvelocity in/minFig. 9. For the high axis transition speed, the measured magnitude ofthe feedrate v along the curve, obtained from the x and y velocitycomponents.S.D. Timar, R.T. Farouki / Robotics and Computer-Integrated Manufacturing 23 (2007) 563579576asymptoticallyapproachestheno-loadspeedastheacceleration decays to zero. This agrees with the speedresponse of a simple rotary inertia to a speed-dependenttorque input presented in 10. As indicated by theplots, the time-optimal feedrate functions are accuratelyrealized by the real-time interpolator algorithms describedin Section 9.11. ClosureThe principles of time-optimal motion control, underprescribed acceleration bounds deriving from physicallimitations of the actuators and controllers, have beenapplied to the problem of computing feedrates for CartesianCNC machines along curved paths. This study treats thecontext of both constant and speed-dependent axis accel-eration bounds, arising from the drive motor current andvoltage ratings. For paths described by polynomial para-metric curves, the time-optimal feedrate is a piecewise-analytic function of the curve parameter. For the curvesegments onwhichconstant accelerationbounds areactive the feedrate can be specified in terms of a rationalfunction of the parameter. For the segments with speed-dependent acceleration bounds, the feedrate expressionincurs a novel transcendental function, that admits accurateand efficient evaluation for use in real-time applications.Real-time CNC interpolator algorithms are presented todrive the machine at the time-optimal feedrate directly fromthe analytic path description, eliminating the need forpiecewise-linear/circular G code path approximations.In practice, depending on the machine actuators andcontrol system, the speed-dependent acceleration regimemay represent a significant portion of the total axisoperating window. Although this complicates the time-optimal feed