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原稿-十字接头零件工艺及车φ22孔夹具设计【4张CAD图纸、工艺卡片和说明书】

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关键词：: 工艺及车φ22孔夹具设计及孔车孔夹具 φ22孔夹具设计 22孔夹具设计车22孔夹具设计十字接头零件十字接头零件工艺及夹具设计十字接头零件十字接头零件夹具设计十字接头零件

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摘要

本次设计的主要内容是十字接头加工工艺规程及孔车孔夹具的设计。十字接头主要是与其它零件配对使用，其主要加工表面及控制位置为内孔和内孔。由零件要求分析可知，保证孔和孔的精度尺寸的同时应该尽量保证其平行度，这对于后工序装配和总成使用上来说都有重要影响。所以，工序安排时，采取以粗定位夹紧，加工内孔后，对孔进行钻扩铰削加工。对于钻2-M10螺孔时，主要以十字接头的四个侧面定位，控制其自由度，以达到加工出来的产品满足要求并且一致性好的目的。本文的研究重点在于通过对十字接头的工艺性和力学性能分析，对加工工艺进行合理分析，选择确定合理的毛坯、加工方式、设计高效、省力的夹具，通过实践验证，最终加工出合格的十字接头零件。

【关键词】十字接头加工工艺夹具设计夹具

Abstract

The connecting rod is one of the main driving medium of diesel engine, this text expounds mainly the machining technology and the design of clamping device of the connecting rod. The precision of size, the precision of profile and the precision of position , of the connecting rod is demanded highly , and the rigidity of the connecting rod is not enough, easy to deform, so arranging the craft course, need to separate the each main and superficial thick finish machining process. Reduce the function of processing the surplus , cutting force and internal stress progressively , revise the deformation after processing, can reach the specification requirement for the part finally .

【Keyword】Connecting rod Deformination Processing technology Design of clamping device

目　录

摘要 3

Abstract 4

第一章绪论 4

第二章零件的分析 5

2.1、零件的作用 5

2.2、零件的工艺分析 5

第三章毛坯的确定以及毛坯图和零件图的绘出 8

3.1、分析毛坯的制作以及其组成 8

3.2、加工余量和工序、毛坯尺寸的确定 8

第四章十字接头的工艺规程的设计 10

4.1、确定定位基准 10

4.2、工艺路线的确定 10

4.3、选择加工设备及刀、夹、量具 14

4.4 确定切削用量及基本工时 16

4.4.1粗精车Φ400.039孔 16

4.4.2 调头车Φ40-Φ53孔左端面 18

4.4.3钻扩铰Φ220.033内孔及左端面,拉Φ220.033内孔油槽 20

4.4.4调头车Φ220.033孔右端面 22

4.4.5铣腰形平面 23

4.4.6钻2-M10螺纹孔底孔2-Φ8.5，钻M6螺纹孔底孔Φ5，钻油孔Φ3,锪孔口φ6深2攻M10，M6螺纹 26

第五章车孔夹具的设计 27

5.1选择定位基准 28

5.2分析误差 29

5.3夹具的夹紧力和切削力的计算 30

5.4夹紧元件及动力装置确定 33

5.5夹具设计及操作的简要说明 34

小结 35

参考文献 36

致谢 37

第一章绪论

本文首先对十字接头的零件进行分析，通过对十字接头进行的研究和分析，描述了十字接头零件的加工工艺的制定，各工序的加工工艺的分析以及切削用量和工时的计算等相关内容。为了提高劳动生产率，降低劳动强度，保证加工质量，与指导老协商后，觉得用夹具比较合适。

在这次毕业设计中，根据课题所给的零件图、技术要求，通过查阅相关资料和书籍，了解和掌握了的机械加工工艺的一般方法和步骤，并运用这些方法和步骤进行了十字接头的机械加工工艺及夹具设计。整个设计的指导思想“简便、高效、经济”。力求生产出符合要求的产品。

第二章零件的分析

2.1 零件的作用

题目所给的零件是十字接头，而十字接头一般是铸件铸造而成，上面钻或铣有直孔或者是螺纹孔，其作用是用来与其它部件进行连接的。同时十字接头零件同时还有转向的作用。一把是用来和有90度转折的零件连接使用的，而本课题的任务就是”十字接头加工工艺及夹具设计“。

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内容简介：: 南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称车工序号10工种车工件等级IT8简图设备名称普通车床材料名称灰口铸铁型号CA6140牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称车40孔夹具编号同时加工零件数1件同时看管机床数1台工时定额基本工时0.51min每件工时定额0.33min辅助工时0.17min准备结束工时0.33min单件工时1.34min每台零件数1附加工时0.33min每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分)工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速r/min每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1粗精车400.039孔内孔车刀游标卡尺1.321703103151.850.50.17工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称车工序号20工种车工件等级IT8简图设备名称普通车床材料名称灰口铸铁型号CA6140牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称弹性芯轴编号同时加工零件数1件同时看管机床数1台工时定额基本工时30s每件工时定额10s辅助工时15s准备结束工时20s单件工时100s每台零件数1附加工时30s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1调头车40-53孔左端面外圆车刀游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称车工序号30工种车工件等级IT8简图设备名称普通车床材料名称灰口铸铁型号CA6140牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称编号车22孔夹具同时加工零件数1件同时看管机床数1台工时定额基本工时40s每件工时定额5s辅助工时10s准备结束工时10s单件工时80s每台零件数1附加工时15s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1粗精车220.033内孔内孔车刀游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称车工序号40工种车工件等级IT8简图设备名称普通车床材料名称灰口铸铁型号CA6140牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称编号弹性芯轴同时加工零件数1件同时看管机床数1台工时定额基本工时35s每件工时定额100s辅助工时20s准备结束工时20s单件工时95s每台零件数1附加工时20s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1调头车220.039内孔右端面外圆车刀游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称铣工序号50工种铣工工件等级IT8简图设备名称普通车床材料名称灰口铸铁型号X52牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称专用夹具编号同时加工零件数件同时看管机床数台工时定额基本工时35s每件工时定额80s辅助工时10s准备结束工时10s单件工时75s每台零件数1附加工时20s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1铣腰形平面立铣刀游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称拉工序号60工种铣工工件等级IT8简图设备名称拉床材料名称灰口铸铁型号L6120牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称拉夹具编号同时加工零件数1件同时看管机床数1台工时定额基本工时55s每件工时定额110s辅助工时30s准备结束工时10s单件工时110s每台零件数1附加工时15s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1拉220.033内孔油槽拉刀游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称钻工序号70工种钻工工件等级IT8简图设备名称钻床材料名称灰口铸铁型号Z525牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称翻转钻模编号同时加工零件数1件同时看管机床数1台工时定额基本工时20s每件工时定额65s辅助工时20s准备结束工时10s单件工时60s每台零件数1附加工时10s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1钻孔3沉孔6，钻M6螺孔底孔5,2-M10螺孔底孔93，,5，,9麻花钻头游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准南昌理工学院机械加工工序卡产品名称十字接头零件名称十字接头共页第页产品型号零件编号No车间机加工段（组）工序名称钻工序号80工种钻工工件等级IT8简图设备名称攻丝机材料名称灰口铸铁型号牌号HT200本厂编号机械性能HB电动功率零件毛重1.14kg冷却液柴油零件净重0.94kg夹具名称攻丝夹具编号同时加工零件数1件同时看管机床数1台工时定额基本工时50s每件工时定额100s辅助工时20ss准备结束工时10s单件工时100s每台零件数1附加工时20s每批零件数1加工尺寸（MM）切削计算长度切削规范工作时间（分工步号工步（位）内容工具规格名称及编号直径、宽长度切深（MM）行刀次数切削速度主轴转速每转每分基本时间辅助刀具量具辅具从至进刀量机动手动MM(转)MM(分)1攻M10，M6螺纹M6，M10丝攻游标卡尺工更序改卡栏通知书栏修改代号数量签字日期通知书写修改代号数量签字日期产更品该图栏更该标记制定校对审核会签批准Robotics and Computer-Integrated Manufacturing 21 (2005) 368378Locating completeness evaluation and revision in fixture planH. Song?, Y. RongCAM Lab, Department of Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USAReceived 14 September 2004; received in revised form 9 November 2004; accepted 10 November 2004AbstractGeometry constraint is one of the most important considerations in fixture design. Analytical formulation of deterministiclocation has been well developed. However, how to analyze and revise a non-deterministic locating scheme during the process ofactual fixture design practice has not been thoroughly studied. In this paper, a methodology to characterize fixturing systemsgeometry constraint status with focus on under-constraint is proposed. An under-constraint status, if it exists, can be recognizedwith given locating scheme. All un-constrained motions of a workpiece in an under-constraint status can be automatically identified.This assists the designer to improve deficit locating scheme and provides guidelines for revision to eventually achieve deterministiclocating.r 2005 Elsevier Ltd. All rights reserved.Keywords: Fixture design; Geometry constraint; Deterministic locating; Under-constrained; Over-constrained1. IntroductionA fixture is a mechanism used in manufacturing operations to hold a workpiece firmly in position. Being a crucialstep in process planning for machining parts, fixture design needs to ensure the positional accuracy and dimensionalaccuracy of a workpiece. In general, 3-2-1 principle is the most widely used guiding principle for developing a locationscheme. V-block and pin-hole locating principles are also commonly used.A location scheme for a machining fixture must satisfy a number of requirements. The most basic requirement is thatit must provide deterministic location for the workpiece 1. This notion states that a locator scheme producesdeterministic location when the workpiece cannot move without losing contact with at least one locator. This has beenone of the most fundamental guidelines for fixture design and studied by many researchers. Concerning geometryconstraint status, a workpiece under any locating scheme falls into one of the following three categories:1. Well-constrained (deterministic): The workpiece is mated at a unique position when six locators are made to contactthe workpiece surface.2. Under-constrained: The six degrees of freedom of workpiece are not fully constrained.3. Over-constrained: The six degrees of freedom of workpiece are constrained by more than six locators.In 1985, Asada and By 1 proposed full rank Jacobian matrix of constraint equations as a criterion and formed thebasis of analytical investigations for deterministic locating that followed. Chou et al. 2 formulated the deterministiclocating problem using screw theory in 1989. It is concluded that the locating wrenches matrix needs to be full rank toachieve deterministic location. This method has been adopted by numerous studies as well. Wang et al. 3 consideredARTICLE IN PRESS/locate/rcim0736-5845/$-see front matter r 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.rcim.2004.11.012?Corresponding author. Tel.: +15088316092; fax: +15088316412.E-mail address: hsong (H. Song).locatorworkpiece contact area effects instead of applying point contact. They introduced a contact matrix andpointed out that two contact bodies should not have equal but opposite curvature at contacting point. Carlson 4suggested that a linear approximation may not be sufficient for some applications such as non-prismatic surfaces ornon-small relative errors. He proposed a second-order Taylor expansion which also takes locator error interaction intoaccount. Marin and Ferreira 5 applied Chous formulation on 3-2-1 location and formulated several easy-to-followplanning rules. Despite the numerous analytical studies on deterministic location, less attention was paid to analyzenon-deterministic location.In the Asada and Bys formulation, they assumed frictionless and point contact between fixturing elements andworkpiece. The desired location is q*, at which a workpiece is to be positioned and piecewisely differentiable surfacefunction is gi(as shown in Fig. 1).The surface function is defined as giq? 0: To be deterministic, there should be a unique solution for the followingequation set for all locators.giq 0;i 1;2;.;n,(1)where n is the number of locators and q x0;y0;z0;y0;f0;c0? represents the position and orientation of theworkpiece.Only considering the vicinity of desired location q?; where q q? Dq; Asada and By showed thatgiq giq? hiDq,(2)where hiis the Jacobian matrix of geometry functions, as shown by the matrix in Eq. (3). The deterministic locatingrequirement can be satisfied if the Jacobian matrix has full rank, which makes the Eq. (2) to have only one solutionq q?:rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0:qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0:qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc026666666664377777777758:9=; 6.(3)Upon given a 3-2-1 locating scheme, the rank of a Jacobian matrix for constraint equations tells the constraint statusas shown in Table 1. If the rank is less than six, the workpiece is under-constrained, i.e., there exists at least one freemotion of the workpiece that is not constrained by locators. If the matrix has full rank but the locating scheme hasmore than six locators, the workpiece is over-constrained, which indicates there exists at least one locator such that itcan be removed without affecting the geometry constrain status of the workpiece.For locating a model other than 3-2-1, datum frame can be established to extract equivalent locating points. Hu 6has developed a systematic approach for this purpose. Hence, this criterion can be applied to all locating schemes.ARTICLE IN PRESSX Y Z O X Y Z O (x0,y0,z0) gi UCS WCS Workpiece Fig. 1. Fixturing system model.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378369Kang et al. 7 followed these methods and implemented them to develop a geometry constraint analysis module intheir automated computer-aided fixture design verification system. Their CAFDV system can calculate the Jacobianmatrix and its rank to determine locating completeness. It can also analyze the workpiece displacement and sensitivityto locating error.Xiong et al. 8 presented an approach to check the rank of locating matrix WL(see Appendix). They also intro-duced left/right generalized inverse of the locating matrix to analyze the geometric errors of workpiece. It hasbeen shown that the position and orientation errors DX of the workpiece and the position errors Dr of locators arerelated as follows:Well-constrained :DX WLDr,(4)Over-constrained :DX WTLWL?1WTLDr,(5)Under-constrained :DX WTLWLWTL?1Dr I6?6? WTLWLWTL?1WLl,(6)where l is an arbitrary vector.They further introduced several indexes derived from those matrixes to evaluate locator configurations, followed byoptimization through constrained nonlinear programming. Their analytical study, however, does not concern therevision of non-deterministic locating. Currently, there is no systematic study on how to deal with a fixture design thatfailed to provide deterministic location.2. Locating completeness evaluationIf deterministic location is not achieved by designed fixturing system, it is as important for designers to knowwhat the constraint status is and how to improve the design. If the fixturing system is over-constrained, informa-tion about the unnecessary locators is desired. While under-constrained occurs, the knowledge about all the un-constrained motions of a workpiece may guide designers to select additional locators and/or revise the locatingscheme more efficiently. A general strategy to characterize geometry constraint status of a locating scheme is describedin Fig. 2.In this paper, the rank of locating matrix is exerted to evaluate geometry constraint status (see Appendixfor derivation of locating matrix). The deterministic locating requires six locators that provide full rank locatingmatrix WL:As shown in Fig. 3, for given locator number n; locating normal vector ai;bi;ci? and locating position xi;yi;zi? foreach locator, i 1;2;.;n; the n ? 6 locating matrix can be determined as follows:WLa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775.(7)When rankWL 6 and n 6; the workpiece is well-constrained.When rankWL 6 and n46; the workpiece is over-constrained. This means there are n ? 6 unnecessary locatorsin the locating scheme. The workpiece will be well-constrained without the presence of those n ? 6 locators. Themathematical representation for this status is that there are n ? 6 row vectors in locating matrix that can be expressedas linear combinations of the other six row vectors. The locators corresponding to that six row vectors consist oneARTICLE IN PRESSTable 1RankNumber of locatorsStatuso 6Under-constrained 6 6Well-constrained 646Over-constrainedH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378370locating scheme that provides deterministic location. The developed algorithm uses the following approach todetermine the unnecessary locators:1. Find all the combination of n ? 6 locators.2. For each combination, remove that n ? 6 locators from locating scheme.3. Recalculate the rank of locating matrix for the left six locators.4. If the rank remains unchanged, the removed n ? 6 locators are responsible for over-constrained status.This method may yield multi-solutions and require designer to determine which set of unnecessary locators shouldbe removed for the best locating performance.When rankWLo6; the workpiece is under-constrained.3. Algorithm development and implementationThe algorithm to be developed here will dedicate to provide information on un-constrained motions of theworkpiece in under-constrained status. Suppose there are n locators, the relationship between a workpieces position/ARTICLE IN PRESSFig. 2. Geometry constraint status characterization.X Z Y (a1,b1,c1) 2,b2,c2) (x1,y1,z1) (x2,y2,z2) (ai,bi,ci) (xi,yi,zi) (aFig. 3. A simplified locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378371orientation errors and locator errors can be expressed as follows:DX DxDyDzaxayaz2666666666437777777775w11:w1i:w1nw21:w2i:w2nw31:w3i:w3nw41:w4i:w4nw51:w5i:w5nw61:w6i:w6n2666666666437777777775?Dr1:Dri:Drn2666666437777775,(8)where Dx;Dy;Dz;ax;ay;azare displacement along x, y, z axis and rotation about x, y, z axis, respectively. Driisgeometric error of the ith locator. wijis defined by right generalized inverse of the locating matrix Wr WTLWLWTL?15.To identify all the un-constrained motions of the workpiece, V dxi;dyi;dzi;daxi;dayi;dazi? is introduced such thatV DX 0.(9)Since rankDXo6; there must exist non-zero V that satisfies Eq. (9). Each non-zero solution of V represents an un-constrained motion. Each term of V represents a component of that motion. For example, 0;0;0;3;0;0? says that therotation about x-axis is not constrained. 0;1;1;0;0;0? means that the workpiece can move along the direction given byvector 0;1;1?: There could be infinite solutions. The solution space, however, can be constructed by 6 ? rankWLbasic solutions. Following analysis is dedicated to find out the basic solutions.From Eqs. (8) and (9)VX dxDx dyDy dzDz daxDax dayDay dazDaz dxXni1w1iDri dyXni1w2iDri dzXni1w3iDri daxXni1w4iDri dayXni1w5iDri dazXni1w6iDriXni1Vw1i;w2i;w3i;w4i;w5i;w6i?TDri 0.10Eq. (10) holds for 8Driif and only if Eq. (11) is true for 8i1pipn:Vw1i;w2i;w3i;w4i;w5i;w6i?T 0.(11)Eq. (11) illustrates the dependency relationships among row vectors of Wr: In special cases, say, all w1jequal to zero,V has an obvious solution 1, 0, 0, 0, 0, 0, indicating displacement along the x-axis is not constrained. This is easy tounderstand because Dx 0 in this case, implying that the corresponding position error of the workpiece is notdependent of any locator errors. Hence, the associated motion is not constrained by locators. Moreover, a combinedmotion is not constrained if one of the elements in DX can be expressed as linear combination of other elements. Forinstance, 9w1ja0;w2ja0; w1j ?w2jfor 8j: In this scenario, the workpiece cannot move along x- or y-axis. However, itcan move along the diagonal line between x- and y-axis defined by vector 1, 1, 0.To find solutions for general cases, the following strategy was developed:1. Eliminate dependent row(s) from locating matrix. Let r rank WL; n number of locator. If ron; create a vectorin n ? r dimension space U u1:uj:un?rhi1pjpn ? r; 1pujpn: Select ujin the way that rankWL r still holds after setting all the terms of all the ujth row(s) equal to zero. Set r ? 6 modified locating matrixWLMa1b1c1c1y1? b1z1a1z1? c1x1b1x1? a1y1:aibiciciyi? biziaizi? cixibixi? aiyi:anbncncnyn? bnznanzn? cnxnbnxn? anyn2666666437777775r?6,where i 1;2;:;niauj:ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 3683783722. Compute the 6 ? n right generalized inverse of the modified locating matrixWr WTLMWLMWTLM?1w11:w1i:w1rw21:w2i:w2rw31:w3i:w3rw41:w4i:w4rw51:w5i:w5rw61:w6i:w6r26666666664377777777756?r3. Trim Wrdown to a r ? rfull rank matrix Wrm: r rankWLo6: Construct a 6 ? r dimension vector Q q1:qj:q6?rhi1pjp6 ? r; 1pqjpn: Select qjin the way that rankWr r still holds after setting all theterms of all the qjth row(s) equal to zero. Set r ? r modified inverse matrixWrmw11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r26666664377777756?6,where l 1;2;:;6 laqj:4. Normalize the free motion space. Suppose V V1;V2;V3;V4;V5;V6? is one of the basic solutions of Eq. (10) withall six terms undetermined. Select a term qkfrom vector Q1pkp6 ? r: SetVqk ?1;Vqj 0 j 1;2;:;6 ? r;jak;(5. Calculated undetermined terms of V: V is also a solution of Eq. (11). The r undetermined terms can be found asfollows.v1:vs:v62666666437777775wqk1:wqki:wqkr2666666437777775?w11:w1i:w1r:wl1:wli:wlr:w61:w6i:w6r2666666437777775?1,where s 1;2;:;6saqj;saqk;l 1;2;:;6 laqj:6. Repeat step 4 (select another term from Q) and step 5 until all 6 ? r basic solutions have been determined.Based on this algorithm, a C+ program was developed to identify the under-constrained status and un-constrained motions.Example 1. In a surface grinding operation, a workpiece is located on a fixture system as shown in Fig. 4. The normalvector and position of each locator are as follows:L1:0, 0, 10, 1, 3, 00,L2:0, 0, 10, 3, 3, 00,L3:0, 0, 10, 2, 1, 00,L4:0, 1, 00, 3, 0, 20,L5:0, 1, 00, 1, 0, 20.Consequently, the locating matrix is determined.WL0013?100013?300011?20010?203010?2012666666437777775.ARTICLE IN PRESSH. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378373This locating system provides under-constrained positioning since rankWL 5o6: The program then calculatesthe right generalized inverse of the locating matrix.Wr000000:50:5?1?0:51:50:75?1:251:5000:250:25?0:5000:5?0:50000000:5?0:526666666643777777775.The first row is recognized as a dependent row because removal of this row does not affect rank of the matrix. Theother five rows are independent rows. A linear combination of the independent rows is found according therequirement in step 5 of the procedure for under-constrained status. The solution for this special case is obvious that allthe coefficients are zero. Hence, the un-constrained motion of workpiece can be determined as V ?1; 0; 0; 0; 0; 0?:This indicates that the workpiece can move along x direction. Based on this result, an additional locator should beemployed to constraint displacement of workpiece along x-axis.Example 2. Fig. 5 shows a knuckle with 3-2-1 locating system. The normal vector and position of each locator in thisinitial design are as follows:L1:0, 1, 00, 896, ?877, ?5150,L2:0, 1, 00, 1060, ?875, ?3780,L3:0, 1, 00, 1010, ?959, ?6120,L4:0.9955, ?0.0349, 0.0880, 977, ?902, ?6240,L5:0.9955, ?0.0349, 0.0880, 977, ?866, ?6240,L6:0.088, 0.017, ?0.9960, 1034, ?864, ?3590.The locating matrix of this configuration isWL010515:000:8960010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:9960866:6257998:24660:093626666666643777777775,rankWL 5o6 reveals that the workpiece is under-constrained. It is found that one of the first five rows can beremoved without varying the rank of locating matrix. Suppose the first row, i.e., locator L1is removed from WL; theARTICLE IN PRESSXZYL3L4L5L2L1Fig. 4. Under-constrained locating scheme.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378374modified locating matrix turns intoWLM010378:001:0600010612:001:01000:9955?0:03490:0880?101:2445?707:26640:86380:9955?0:03490:0880?98:0728?707:26640:82800:08800:0170?0:996866:6257998:24660:09362666666437777775.The right generalized inverse of the modified locating matrix isWr1:8768?1:8607?20:666521:37160:49953:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775.The program checked the dependent row and found every row is dependent on other five rows. Without losinggenerality, the first row is regarded as dependent row. The 5 ? 5 modified inverse matrix isWrm3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:028402666666437777775.The undetermined solution is V ?1; v2; v3; v4; v5; v6?:To calculate the five undetermined terms of V according to step 5,1:8768?1:8607?20:666521:37160:499526666666643777777775T?3:0551?2:0551?32:444832:44480?1:09561:086212:0648?12:4764?0:2916?0:00440:00440:0061?0:006100:0025?0:00250:0065?0:00690:0007?0:00040:00040:0284?0:0284026666666643777777775?1 0; ?1:713; ?0:0432; ?0:0706; 0:04?.Substituting this result into the undetermined solution yields V ?1;0; ?1:713; ?0:0432; ?0:0706; 0:04?ARTICLE IN PRESSFig. 5. Knuckle 610 (modified from real design).H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378375This vector represents a free motion defined by the combination of a displacement along ?1, 0, ?1.713 directioncombined and a rotation about ?0.0432, ?0.0706, 0.04. To revise this locating configuration, another locator shouldbe added to constrain this free motion of the workpiece, assuming locator L1was removed in step 1. The program canalso calculate the free motions of the workpiece if a locator other than L1was removed in step 1. This provides morerevision options for designer.4. SummaryDeterministic location is an important requirement for fixture locating scheme design. Analytical criterion fordeterministic status has been well established. To further study non-deterministic status, an algorithm for checking thegeometry constraint status has been developed. This algorithm can identify an under-constrained status and indicatethe un-constrained motions of workpiece. It can also recognize an over-constrained status and unnecessary locators.The output information can assist designer to analyze and improve an existing locating scheme.Appendix. Locating matrixConsider a general workpiece as shown in Fig. 6. Choose reference frame fWg fixed to the workpiece. Let fGg andfLig be the global frame and the ith locator frame fixed relative to it. We haveFiXw;Hw;rwi fiXli;Hli;rli,(12)where Xw2 3?1and Hw2 3?1(Xli2 3?1and Hli2 3?1) are the position and orientation of the workpiece(the ith locator) in the global frame fGg; rwi2 3?1(rli2 3?1) is the position of the ith contact point between theworkpiece and the ith locator in the workpiece frame fWg (the ith locator frame fLig).Assume that DXw2 3?1(DHw2 3?1) and Drwi2 3?1are the deviations of the position Xw2 3?1(orientationHw2 3?1) of the workpiece and the position of the ith contact point rwi2 3?1; respectively. Then we have the actualcontact on the workpiece,FiXw DXw;Hw DHw;rwi Drwi FiXw;Hw;rwi qFiqXwDXwqFiqHwDHwqFiqrwiDrwi,13where the second term in the right side of Eq. (13) is the position error of the ith contact point resulting from theposition error DXwof the workpiece, the third term is the position error of the ith contact point resulting from theorientation error DHwof the workpiece, and the fourth term is the position error of the ith contact point resulting fromits workpiece geometric variation Drwion the workpiece.Similarly, assume that DXli2 3?1DHli2 3?1 and Drli2 3?1are the deviations of the position Xli2 3?1(orientation Hli2 3?1) of the ith locator and the position of the ith contact point rli2 3?1; respectively. The actualcontact on the ith locator isfiXli DXli;Hli DHli;rli Drli fiXli;Hli;rli qfiqXliDXliqfiqHliDHliqfiqrliDrli,14ARTICLE IN PRESSworkpieceith locator Win iLiwrilr GilXwXwX wFig. 6. Fixture coordinate frames.H. Song, Y. Rong / Robotics and Computer-Integrated Manufacturing 21 (2005) 368378376where the second term in the right side of Eq. (14) is the position error of the ith contact point resulting from theposition error DXliof the ith locato

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