三自由度并聯(lián)機器人的結構設計與運動學分析【說明書+CAD+SOLIDWORKS】,說明書+CAD+SOLIDWORKS,三自由度并聯(lián)機器人的結構設計與運動學分析【說明書+CAD+SOLIDWORKS】,自由度,并聯(lián),機器人,結構設計,運動學,分析,說明書,CAD,SOLIDWORKS 機械手的給定工作區(qū)內(nèi)的一種 6 自由度并聯(lián)關鍵點三維設計方法 摘要：本文提出了在給定工作區(qū)內(nèi)一種6自由度的新三維設計新方法 。許多關鍵特性已經(jīng)進行運動學分析和拉格朗日乘數(shù)法。此外，在整個機械手的直接幾何關系中導出了參數(shù)。提出了設計方法，關于這些關鍵點特性具有很高的效率和準確性。此外，避免了復雜機械手的工作空間和無量綱化推導分析從而可能讓這種方法的廣泛應用。 ? 2014年愛思唯爾有限公司。版權所有 1 .導言 對并聯(lián)機器人的關注主要是發(fā)現(xiàn)他們有更好的承載能力，更好的剛度，和比串聯(lián)機器人更好的精度[1-4]。因此并聯(lián)機器人的研究已成為一個熱門的國際機器人研究領域?[5-9]。并聯(lián)機器人的設計過程是機械產(chǎn)品中最具有挑戰(zhàn)性的問題。設計機器人[10-12]的配置，機械臂的幾何參數(shù)應由三維設計決定。引用?[13，14]?中提出的參數(shù)設計方法分別用于6 自由度歌賦型機器人和3自由度并聯(lián)機器人。 一般來說，最重要的設計目標之一是讓機器人在給定工作區(qū)工作。到目前為止，有主要有兩種方法，根據(jù)給定的工作區(qū)的并聯(lián)機器人的幾何參數(shù)優(yōu)化設計。第一次使用多點來描述給定工作區(qū)，然后檢查機械手的每個點的設計要求是否符合參數(shù)[15-17]，與另一個邊界的機械手之間建立參數(shù)和工作區(qū)中的函數(shù)，然后確保給定工作區(qū)是機械手的工作空間邊界內(nèi)[18-22]。基于我們在這項研究發(fā)現(xiàn)的幾個關鍵問題，本文試圖探索給定的工作區(qū)6自由度并聯(lián)機器人新的三維設計方法。這種設計方法是快速的，它的結果是準確的。 在我們以前的工作中，這種新型的6自由度并聯(lián)機構中用到了?3-3'-PSS?配置。 與傳統(tǒng)?6-SPS?并聯(lián)機器人相比這?3-3'-PSS?并聯(lián)的機械臂性能允許更高的各向同性的、更大的旋轉范圍移動平臺，減少了身體慣性。 若要開始設計，應清楚的描述所需的工作區(qū)。因為不能以圖形方式表示?6 維工作區(qū)，以人類可讀的方式，沒有一般的方法來分析確定的?6-D?工作區(qū)的邊界6 自由度并聯(lián)機器人，大多數(shù)文獻?[23-27]?將?6-D?區(qū)劃分為工作區(qū)的位置和方位工作空間。工作區(qū)的位置是指機械手的移動平臺可以達到一定的取向的空間。它可以容易地描述。方位工作空間是移動平臺可以實現(xiàn)在某一時刻的所有方向的集合。然而，由于旋轉角度的復雜性，方位工作空間很難確定和代表?？紤]到我們并聯(lián)機械手的對稱性，簡明描述?6-D?區(qū)找到了種的三維設計。 本文的結構如下。第二節(jié)介紹了建模的設計問題及運動學分析。第3節(jié)介紹如何找到關鍵點特征。第4節(jié)中，討論了設計方法及應用。最后，第5節(jié)中總結發(fā)言。 2．模型的設計問題和力學分析 新的PSS '3-3并聯(lián)機器人的結構如圖1所示，它是由一個移動的平臺，一個固定基座， 和六個具有相同的幾何結構支撐臂組成。四肢編號從1到6的每個肢體由一個棱柱形接頭，一個球形接頭和聯(lián)合空間綜合信息網(wǎng)絡球系列連接到固定基地到所述移動平臺。一個線性執(zhí)行機構驅動的棱柱沿著固定軌道各肢的關節(jié)。關節(jié)Bi和關節(jié)Ai之間是長為Li的剛性連桿（I =1，...，6） 1，2，和3被設置成位于一水平面的PB它們的軸線四肢的三個線性致動器，且當這些軸不交于一點時它們的軸之間的夾角為120°。這些軸與操縱器的對稱軸之間的距離是相同的，在這里我們使用一個參數(shù)來表示該距離。其他三個線性執(zhí)行器四肢4,5，和6被設置成垂直的軸線。關節(jié)的移動平臺A1?A6分布在中心對稱的半徑為a的一個圓上。這種操縱器的中心在平面PB的交叉點和操縱器的對稱軸上，在其上連接有固定笛卡爾參考幀-O {X，Y，Z}。固定框架y軸和z軸都在平面PB上，并且與操縱器的對稱軸的X軸重合。移動框架O'{X'，Y'，Z'}連接移動平臺O點“，這是指向位于圓心上的A1?A6。關于機械手是軸對稱的事實，移動臺處于初始位置時讓點O'與點O重合，從而操縱器的工作空間相對于固定框ò也是軸對稱。 設計的操縱器的幾何參數(shù)前，所需的工作空間應明確說明。從前面的討論中可以看出，簡明地描述所需6-D的工作區(qū)是一個具有挑戰(zhàn)性的問題。在這個研究中，對移動臺的方向的說明，僅指示向量（顯示在圖2中），而不是繞其對稱軸旋轉而言。事實上，這是許多機床有著的同樣的情況。在此基礎上，我們使用一組特殊的歐拉角來表示的移動平臺的方向。移動平臺的首先由一個角度φ固定x軸，然后由角度θ固定z軸，最后由角φ固定x軸（圖2）。我們可以把旋轉矩陣簡單的寫成這種情況： 3.在給定的工作空間機器人的關鍵特征 在這項研究中，通過大量的計算，我們發(fā)現(xiàn)在qi最大范圍內(nèi)，盡管給定的工作區(qū)和操縱器的尺寸在改變，αBi和αAi總是發(fā)生在一定位置。這一特點對尺寸設計非常有幫助，所以我們稱這些位置為關鍵點。本節(jié)將證明理論上使用拉格朗日乘子的方法，建立關鍵點。 為了推廣，我們做了三維設計的相關參數(shù)量通過讓他們每個人用鋼筋混凝土進行劃分。因此，工作空間汽缸的無量綱半徑為1，并且其無量綱高度為2H。其中，H= HC / Rc。因此，基于該無量綱工作空間的尺寸設計的結果不能被直接當作操縱器的幾何參數(shù)，除非由RC乘以它們所有（應當注意的是，在此過程中角度不影響）。由于機械手的配置兩肢體的人群有不同的關鍵特征。因此，兩肢組的特性，應分別研究。 4.基礎的三維設計方法的關鍵點及其應用 在上一節(jié)找到對應的工作空間內(nèi)操縱的一些重要關鍵點的特征。其要點是極端位置，這將導致在給定的工作空間中操縱器的最壞運動學條件。如果操縱器可在關鍵點達到所需的運動學性能，那么這個運動性能將在給定的工作空間中保證每個點。這些特性可以被用于確定所述操縱器的幾何參數(shù)，從而在三維設計將具有非常高的效率和準確性。對于這個關鍵點的設計方法的主要步驟如下： 1.描述所需的工作空間。研究了操縱器的工作任務，并計算出所需要的空間和方向。然后選擇與可以只達到要求的給定的工作空間有一定指向靈巧指數(shù)缸。如果所需的工作空間是復雜的，它可以被描述為多個同軸圓柱體具有不同指向靈巧指數(shù)與圖4所示。在這種狀態(tài)下，下面的設計步驟2-5，對于每個氣缸都應進行，其結果應結合作為最終的解決方案。 2.給定的工作空間量綱。對于每個氣缸，讓其半徑和高度由它自己的半徑進行劃分。 3.明確額外的設計要求和使用表1中找到所有需要的關鍵點。如果關節(jié)角的范圍沒有限制，可以與工作區(qū)保證的關鍵點或相應的直接關系建立所述幾何參數(shù)的約束關系。（參考表1）。如果接頭角度是有要求限制的，應與最大αBi和最大αAi的關鍵點或相應的直接關系建立所述幾何參數(shù)的約束關系。（參考表 1） 4.確定的幾何參數(shù)。找到能滿足前面建立的步驟中的約束關系的適當?shù)膮?shù)。這些約束關系，a和Li有許多可能的解決方案可以找到。一般最小的a和Li將導致操作者的最小量應被選擇。應當注意的是，只有一個肢需要被確定給每個組，因為操作者是對稱。在一些情況下，a和Li可能有具有因工作任務的額外的限制，并且步驟可用于進一步優(yōu)化設計的約束關系。 5.獲得的a和Li應應乘以圓柱的半徑得到維數(shù)。然后他們可以作為機器人的幾何參數(shù)。 6.確定其余的幾何參數(shù)。 如果有多于一缸用于工作區(qū)說明，在第?5?步中得到的結果應該作為一個相結合 解決方案。那就是，選擇的最大值和李之間所有氣缸的結果作為最后的解決辦法。因此，聯(lián)合解決方案： 能滿足各種約束關系的每個氣缸。在那之后，αBi?和?αAi?的范圍應當重新計算與最終解決方案的關鍵點船帆齊和最低氣或?（請參閱表?1）?的直接對應關系，可以確定李和練習場。應當指出：所有氣瓶必須檢查在此過程中，其結果應作為最后的結果相結合。 在這里，我們的項目用來證明該設計方法的應用。我們所需的工作區(qū)可以用描述 筒?（缸?1）?與半徑為?600?毫米，高度為?800?毫米和?0 °?時，指向靈巧和氣缸（缸?2）?與半徑 200?毫米、?高度為?400?毫米和?30 °?的指點靈巧。各關節(jié)角度被限制為小于?45 °。此外，參數(shù) 需求大于?350?毫米將在移動平臺放置對象的尺寸和接頭的尺寸。為缸?1，與最大值?αBi?和最大值?αAi?的關鍵點，可以獲得參數(shù)的最小的解作為Li=1050?毫米?（i?=?1 2、 3）?和Li?=?850?毫米?（i?=?4,5,6）?而不是參與。油缸2，最小的解的參數(shù)可以作為發(fā)現(xiàn)?a=?350?毫米，Li=?1050?毫米?（i?=?1 2、 3）?和Li=?1000?毫米（i=?4,5,6）?與要點船帆?αBi?和最大值?αAi。結合這兩項結果，可以得到該機械手的最終解，作為?a=?350?毫米，Li=?1050?毫米?（i= 1 2、 3）?和Li=?1000?毫米?（ia=?4,5,6）。最后，為每個氣缸帶有計算的?αBi、αAi?和駕駛中風最后的范圍相應的關鍵點，然后結合。設計結果如表?2?所示。和與該機械手的原型 這些設計的幾何參數(shù)如圖?5?所示。 為了驗證這些設計結果的正確性，設計的機械手性能在給定工作區(qū)中有 已檢查。我們采取了一系列圓筒截面和離散他們成均勻離散點。每個這些離散點的取向也進行離散化處理。 然后聯(lián)合角度的值記錄在移動平臺達到每個位置和方向。 為清楚起見，都會選擇一些典型的數(shù)據(jù)并繪制在這部分中。當設計的機械手工作缸?2?頂塊、?分布的?αBi?和?αAi組?1?所示圖?6?和?7分別。圖?8?和圖?9?顯示了同樣的情況，αBi?和?αAi?2?組。可以觀察到所有關節(jié)角度都小于45 °，并只是接近?45 °?腿各關節(jié)角度的最大值出現(xiàn)在的關鍵點。所有這些結果都是一致的。 本文分析研究并滿足要求 5.結論 本文對此提出了新的三維設計方法，為我們的新?' 3-3'-PSS?并聯(lián)機構根據(jù)給定提出了工作區(qū)。這種方法基于幾個關鍵點，避免了機械手的復雜分析自己?6-D?區(qū)實際上并沒有一個統(tǒng)一的描述人類可讀的方式。關鍵點建立簡單的關系機械臂的幾何參數(shù)與工作區(qū)的要求。在此基礎，提出的設計方法已 非常高的效率和準確性。 很多關鍵點特征已發(fā)現(xiàn)并在表?1?中列出。 要點是極端的立場,將導致最嚴重的機械手的運動學條件給定的工作區(qū)。運動學性能可以保證在整個工作區(qū),讓機械手實現(xiàn)性能的關鍵點。此外,一些直接運動學和幾何參數(shù)之間的關系已經(jīng)建立的空間設計。簡明地描述6 d工作區(qū),使設計要求很明顯,已經(jīng)發(fā)現(xiàn)了對稱描述給定的工作區(qū)。這個描述很容易理解和接近機械手的操作條件。因此,這種方法可以很容易地用在許多不同的情況。關鍵點是會導致極端的立場。 機械手在給定工作區(qū)中的最差運動學條件。運動學性能可以保證內(nèi) 給定工作區(qū)，讓整個機械手實現(xiàn)性能的關鍵點。此外，一些直接的關系 之間的運動學和幾何參數(shù)已經(jīng)被為三維設計建造。 簡要描述?6-D?區(qū)和清楚的設計要求，對稱的描述找到了給定工作區(qū)。此描述是機械手的非常容易理解和接近工況。其結果是，這種方法可輕松用于許多不同的情況。這種方法推導了特定類型的并行機制，但找到關鍵點的想法可能會用于其它并聯(lián)機構的類型。核心問題是找到其職位訂明的工作區(qū)中是獨立的關鍵點。 隨著規(guī)模的訂明的工作區(qū)和機制。這通常需要訂明的工作區(qū)的形狀和機制的工作區(qū)有一些相似的特征如本例中的軸向對稱。在此研究中，任何其他軸對稱的形狀可以用于描述形狀的除了氣缸的給定工作區(qū)。重寫的約束方程拉格朗日方法，以及這些形狀的關鍵點，可以發(fā)現(xiàn)與本文類似的程序。可能很難找到關鍵點，但三維設計的并行機制會變得非常方便一旦它做了。如果機制是不對稱的,那么它應當指出的關鍵點應分別為每個肢體找到。 提出的設計方法基于運動學。其實，關節(jié)角?αBi?和?αAi，本文主要研究有直接 雅可比矩陣，然后動態(tài)的關系?；谶@項工作，在不久的將來，將研究基于動力學的設計方法。 確認 這項工作部分支持主要國家基本研究中國的發(fā)展計劃?（973?計劃） （第?2013CB035501?號），和國家自然科學基金?（批準號：?51335007）。 文獻資料 [1] B. 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A key point dimensional design method of a 6-DOF parallelmanipulator for a given workspaceRui Cao1, Feng Gao,1, Yong Zhang1, Dalei Pan1State Key Lab of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, PR Chinaa r t i c l ei n f oa b s t r a c tArticle history:Received 3 April 2014Received in revised form 7 November 2014Accepted 8 November 2014Available online 25 November 2014This paper presents a new method of dimensional design for a 6-PSS parallel mechanismaccording to a given workspace. A symmetrical description has been found to describe the 6-Dworkspace concisely for the dimensional design. Many key point characteristics have beenfound and verified by the kinematic analysis and the method of Lagrange multipliers.Furthermore,thedirectrelationsbetweenthegivenworkspaceandthemanipulatorsgeometricalparameters have been derived. The proposed design method which is based on these key pointcharacteristics has very high efficiency and accuracy. Additionally, the avoiding of the complexanalysis of the manipulators workspace and the dimensionless derivation make the possibilityof wide use of this method. 2014 Elsevier Ltd. All rights reserved.Keywords:Parallel manipulatorDimensional designWorkspace6-PSSKey point1. IntroductionThe interest for parallel manipulators arises from the fact that they have better load-carrying capacity, better stiffness, and betterprecision than serial manipulators 14. Thus the research on designing parallel manipulators has become a hot topic in theinternational robotic research area 59. The design of parallel manipulators is a challenging problem in the machinery productdesign process. The type synthesis is for designing the configuration for manipulators 1012. And then the geometrical parametersofmanipulatorsshouldbedetermined bythedimensionaldesign.Becausethetypesof parallelmechanisms arealmostunlimited,thedimensionaldesignmustbebasedonacertaintypeofmechanisms.Theparameterdesignmethodspresentedinreference13,14arebased on 6-DOF Gough-type manipulators and 3-DOF parallel manipulators, respectively.Generally, one of the most important design objectives is to let the manipulator work in a given workspace. Therefore, thedimensional design of parallel manipulators for a given workspace is an important problem, which has not gained too much interest.So far, there are mainly two ways to design the geometrical parameters of parallel manipulators according to a given workspace. Thefirst one uses many points to describe the given workspaceand then check whether the manipulator with certain parameters fits thedesign requirements at each point 1517. The other one establishes a function between the parameters and the workspaceboundaries of the manipulator, then make sure that the given workspace is within the manipulators workspace boundaries1822.Basedonseveralkeypointsthatwehavefoundinthisstudy,thispaperattemptstoexploreanewwayofdimensionaldesignfor a new 6-DOF parallel manipulator according to a given workspace. This design method is fast and its result is accurate.In our previous work, a new type of 6-DOF parallel mechanism with an orthogonal 3-3-PSS configuration has been proposed.Compared with the traditional 6-SPS parallel manipulators, this 3-3-PSS parallel manipulator allows higher isotropy of themanipulators performance, larger rotation range of the moving platform and less body inertia.Mechanism and Machine Theory 85 (2015) 113 Corresponding author.E-mail addresses: (R. Cao), (F. Gao), (Y. Zhang), (D. Pan).1P.O. Box ME290, Mechanical Building, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, PR China.http:/dx.doi.org/10.1016/j.mechmachtheory.2014.11.0040094-114X/ 2014 Elsevier Ltd. All rights reserved.Contents lists available at ScienceDirectMechanism and Machine Theoryjournal homepage: workspace refers to a space that the manipulators moving platform can reach with a certain orientation. And it can be easilydepicted.Theorientationworkspaceisthecollectionofalltheorientationsthatthemovingplatformcanachieveatacertainpoint.How-ever, due to the complexity of the rotating angles, the orientation workspace is difficult to be determined and represented. Consideringthe symmetry of our parallel manipulator, a concise way of describing the 6-D workspace is found for the dimensional design.The paper is organized as follows. Section 2 introduces the modeling of the design problem and the kinematics analysis. Section 3shows how the key point characteristics are found. The design method and its application are discussed in Section 4. Finally,concluding remarks are presented in Section 5.2. Modeling of the design problem and kinematic analysisThearchitectureofthenew3-3-PSSparallelmanipulatorisshowninFig.1,whichiscomposedofamovingplatform,afixedbase,and six supportinglimbswith identical geometrical structure. The limbs are numbered from 1 to 6. Each limbconnects the fixed basetothemovingplatformbyaprismaticjoint,asphericaljointBiandasphericaljointAiinseries.Alinearactuatoractuatestheprismaticjoint of each limb along a fixed rail. Between the joint Biand joint Aiis a rigid link of length Li(i=1,6).The three linear actuators of the limbs 1, 2, and 3 are arranged with their axes located in a horizontal plane PB, and the angles be-tween each of their axes are 120 while these axes do not intersect at one point. The distances between these axes and the symmetryaxis of the manipulator are the same, and here we use the parameter a to represent this distance. The other three linear actuators ofthe limbs 4, 5, and 6 are arranged with their axes vertically. The centers of the joints A1 A6of the moving platform are distributedsymmetrically on a circle of radius a. The center of this manipulator is at the intersection of the plane PBand the symmetry axis ofthe manipulator, on which attached a fixed Cartesian reference coordinate frame Ox, y, z. The fixed frames y-axis and z-axis are inthe plane PB, and its x-axis coincides with the symmetry axis of the manipulator. A moving frame O x , y , z is attached on themoving platform at point O which is the center of the circle that points A1 A6located on. Considering the fact that the manipulatoris axisymmetric, let point O coincides with point O when the moving platform is at the initial position. Thus the workspace of themanipulator is also axisymmetric with respect to the fixed frame O.Before designing the geometrical parameters of the manipulator, the required workspace should be clearly described. As can beseen from the previous discussion, concisely describing the required 6-D workspace is a challenging problem. In this research, forthe orientation description of the moving platform, only the pointing vector (showed in Fig. 2) rather than the rotation about itssymmetry axis is concerned. In fact this has the same situation for many machine tools. Based on this, we use a special set of Eulerangles to represent the orientation of the moving platform. The moving platform first rotates about the fixed x-axis by an angle-,thenaboutthefixedz-axisbyanangle,andfinallyaboutthefixedx-axisbytheangle(Fig.2).Andwecansimplywritetherotationmatrix for this case as:R Rot x;Rot z;Rot x;ccssscss2 c2cs sccsscs ccsc2 s2c2435;1Fig. 1. The configuration of the proposed 3-3-PSS parallel manipulator.2R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113where c stands for cosine, s stands for sine, , and 0, , respectively. For the convenience of description, Eq. (1) can beabbreviated asR r11r12r13r21r22r23r31r32r332435:2And it can be observed thatr32 r23sin 2 1cos 2:3This special set of Euler angles gives anintuitive representation of themovingplatforms orientation.The pointingvector is decid-ed byand . Duetothesymmetryof themanipulator, itis easytofindoutthat therange ofis unlimitedwhile isnot.Thus,all thepossiblepointingvectorsthatthemovingplatformcanachieveatacertainpointconstituteacone.Andtheapertureoftheconeisonlyrelated to the maximum range of which is represented by m. We call mas the pointing dexterity index of the moving platform.To take advantage of the symmetry of the manipulator, we restrict the required workspace as a symmetric space. Hence, wedescribethegivenworkspaceasacylinderwithradiusofRc,andheightof2Hc.Additionally,themanipulatorshouldhavethepointingdexterityof mat any pointwithin this cylinder. This human-readable workspacedescription fitsforthe manipulators symmetry andmakes the design objective clearly. Knowing that this workspace description is actually 5-DOF, to represent a 6-DOF workspace, anadditionaldexterityindexoftherotationaboutthemovingplatformssymmetryaxis isneeded.In thissituation,themovingplatformshould first perform an additional rotation about the fixed x-axis by an angle , and the rotation matrix can be written as Rot(x,)Rot(z,)Rot(x, -)Rot(x,). However, 5-DOF is enough for our current study and most multi-DOF machine tools.After the analysis of the required workspace, what parameters of the manipulator need to be determined should be clarified. Thefollowing part will find this out by analyzing the kinematics of the manipulator. As the six limbs of the manipulator have identicalgeometrical structure, we can choose one typical limb for the analysis and its vectors are described in Fig. 3. The linear actuatorsaxis is represented by eiwhich is a unit vector. The direction of the rigid link is represented by liwhose magnitude is Li. The vectorbetween O and the center of the joint Aiis represented by ai with respect to the moving frame O , and aiwith respect to thefixed frame O. It can be found from the previous part that the magnitude of ai/aiis a. When the manipulator at the initial positionthatmentionedabove,ei(i= 1,2,3)isperpendiculartoai,itshouldbenoted.AndtheinitialpositionofBiinthissituationisrepresent-ed by point Ciwhose position vector is ci. With the special set of Euler angles, the transformation from the moving frame to the fixedframe can be described by the position vector of the moving platform p = PxPyPzT, and the rotation matrix R. Thus the generalizedcoordinates of the moving platform can be described as (Px, Py, Pz, , , 0).Let qirepresent the stroke of the linear actuator. Then we can simply get the following relation from Fig. 3:li p Ra0iqieici:4In some cases, the joints Biand Aiwhose stiffness are the lowest of the manipulator need a strong structure to increase theirstiffness. However, the strong structure always limits the rotation ranges of these joints. Therefore, the swing amplitude of theFig. 2. The pointing dexterity and the special set of Euler angles.3R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113rigid link should be studied. We define the angle between liand eias the joint angle Biof joint Bi. As joint Aiis fixed on the movingplatform,thedefinitionshouldwithrespecttothemovingframeOx,y,z.ThusthejointangleAiofjointAiisdefinedastheanglebetween liand Rei. Biand Aiare depicted in Fig. 3. The following equations about Biand Aican be achieved by their defini-tion:li? ei LicosBi5li? Rei LicosAi:6According to these definitions, the rotation of the rigid link about its own axis liis not involved. So Biand Airepresent the swingamplitudeoftherigidlinkwithrespecttotheconnectingjoint.ThemaximumvaluesofBiandAiareveryimportantforthedesignofthe spherical joints and meaningful for avoiding the interference between the rigid links.The six limbs canbedividedintotwogroupsaccordingtotheconfiguration ofthemanipulator.Thelimbs1,2,and3are containedin group 1, and the limbs 4, 5, and 6 in group 2. These two groups have different kinematic characteristics, thus need to be studiedseparately. For the sake of symmetry, the rigid links in one group should have the same length. In group 1 for i = 1, 2 and 3, afixed Cartesian reference coordinate frame Oaix, y, z is attached at the point O. For simplicity and without losing the generality,we let its y-axis point in the negative direction of the vector eiand let its x-axis coincide with the x-axis of the frame Ox, y, z.With respect to the frame Oai, it can be known from the architecture of the manipulator that ei 010?T, a0i 00a?Tand ci 0Lia?T. Assume that lilxlylz?T. Substituting all the known variables into Eq. (4) yields the followingequations:lx ar13 px7ly ar23 pyLi qi8lz ar33 a pz:9Furthermore, the following relation can be achieved with the fact that Liis the magnitude of vector li:ly ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zq:10According to Eq. (10), lyhas two possible solutions. When a coordinate of the moving platform makes Li2 lx2 lz2b 0, lyhas nosolution, which means that this coordinate is out of the manipulators reachable workspace. The situation Li2 lx2 lz2= 0 meansthat the moving platform reaches the boundary of the reachable workspace. This situation is singular and should be avoided inFig. 3. One typical limb of the manipulator.4R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113practice. Because of these, the sign of lyshould be constant during the operation of the manipulator. Let p = 0 and R = I when themoving platform at the initial position. Substituting them into Eq. (8) yieldsly a ? 0 0Li0 Lib0:11Therefore, Eq. (10) should take a negative sign. Then substitute Eq. (10) into the left side of Eq. (8) and we can get the inverse so-lution of the actuating stroke qiof group 1qi ar23py LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 ar13 px2 ar33 a pz2q12lican be written with Eq. (7), Eq. (9), and Eq. (10) asliar13 pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2xl2zqar33 a pz264375:13Then we can obtainli? eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar13 px2 ar33 a pz2q;14li? Rei my mx mz:15Where, my= r22li ei, mx= ar12r13+ r12pxand mz= ar32r33+ r32a + r32pz.In each limb of group 2 (i = 4, 5, 6), for simplicity and without losing the generality, a fixed Cartesian reference coordinate frameOaix, y, z is also attached at the point O with its z-axis intersecting eiand its x-axis coinciding with the x-axis of the frame Ox, y, z.Hence, it can be observed from the architecture of the manipulator thatei 100?T,a0i 00a?Tandci Li0a?Twith respect to the frame Oai. Substituting all the known variables into Eq. (4) yields the following equations.lx ar13 pxLi qi16ly ar23 py17lz ar33 a pz18and the following relation can also be obtained with the fact that Liis the magnitude of vector lilx ?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2yl2zq:19Similartotheanalysis of group 1, we can obtain that Eq. (19) should take a negative sign.Substitute it into the leftside of Eq. (16),and we can get the inverse solution of qifor group 2.qi ar13px LiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar23 py?2 ar33 a pz2r:20With Eq. (16), Eq. (17), and Eq. (18), lican be written asliffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2il2yl2zqar23 pyar33 a pz264375:21Then we can obtainli? eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2i ar23 py?2 ar33 a pz2r;22li? Rei m2x m2y m2z?:235R. Cao et al. / Mechanism and Machine Theory 85 (2015) 113Here, m2x= r11li ei, m2y= ar21r23+ r21pyand m2z= ar31r33+ r31a + r31pz.Through the analysis above, it can be found that among the manipulators geometrical parameters, only a and Liare independentand need to be determined. And the maximum ranges of qi, Bi, and Aineed to be found out for the manufacture of the manipulator.Thus the design problem can be summarized as follows.Requirements:1. The manipulator should achieve the given workspace which is a cylinder with radius of Rc, and height of 2Hc. And the manip-ulator should have the pointing dexterity of mat any point of the given workspace. In some cases, when the requiredworkspace is complicated, the given workspace can be described as many coaxial cylinders with different pointing dexterityindices as Fig. 4 shows.2. Insomesituations,themaximumrangesofBiandAiarelimitedforthepurposeofincreasingthestiffnessofthejointsoravoidingthe interference.Design task:1. Find out appropriate geometrical parameters of a and Lithat can let the manipulator meet with all the requirements listed above.2. Afterthedeterminationofthegeometricalparametersabove,findoutthemaximumrangesofqi,Bi,andAiforthemanufactureofthe actuators and joints.3. The key point characteristics of the manipulator within the given workspaceIn this research, though a large amount of calculations, we have found that the maximum ranges of qi, Bi, and Aialwaysoccur at some certain locations in spite of the dimension changing of the given workspace nor the manipulator. In otherwords, there are some certain relations between these locations and the given workspace. This characteristic is very helpfulfor the dimensional design, thus we call these locations as key points. This section will prove the existence of these key pointstheoretically using the method of Lagrange multipliers and establish the relations between the key points and the givenworkspace.Forthesakeofgeneralization,wemaketherelatedparametersofthedimensionaldesigndimensionlessbylettingeachofthembedivided by Rc. Thus the workspace cylinders dimensionless radius is 1, and its dimensionless height is 2H. Where H = Hc/Rc. As a re-sult, the results of the dimensional design based on this dimensionless workspace cant be treated as the geometrical parameters ofthe manipulator directly, unless multiply each of them by Rc(it should be noted that the angles are not affected in this procedure).The two limb groups have different key point characteristics due to the configuration of the manipulator. So the characteristics ofthe two limb groups should be studied separately.Fig. 4. A description of the given workspace.6R. Cao et al. / Mechanism and Machine Theory 85 (2015) 1133.1. Group 1 (i = 1,2,3)As the given workspace is a cylinder, the points in the given workspace must meet the following equations:p2y p2z1;24HpxH:25From theanalysis in Section 2, we know that lyinEq. (10)should have a solution to let themanipulator reach the current positionand orientation. So it can be derived thatL2iN ar13 px2 ar33 a pz2:26The maximum value of therightpart of Eq. (26) represented by is studied. Andthe parameters involved are px, pz, and . Theirconstraint equations can be written asg1 p2y p2z1g2 pxHg3 pxHg4 mg5 8:27Thus the Lagrange function can be written as 1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:28The extreme values of occur where the gradient of K is zero. The partial derivatives arepx 0;py 0; 0; 0k 0;k 1;5k 2kk 0;k 1;58:29By solving the equation system (Eq. (29) and comparing the extreme values, we can obtain that reaches its maximum valuewhen px H;pz 1; 2; m, or px H;pz 1; 2; m. Substituting these two solutions into Eq. (26) yieldsL2iN asm H2 acm a 12:30Toensurethatthemanipulatorcanreacheverypointofthegivenworkspace,aandLishouldbechosentolettheEq.(30)establish.Andthegeneralizedcoordinateofthekeypointsforthiscaseare H;1;2;m;0?and H;1;2;m;0?.Where meansthis valueis arbitrary.Inordertofindthemaximumrangeofqi,weneedtofindboththeminimumandmaximumvaluesofqi.AndtheLagrangefunctionfor finding its minimum value can be written as qi1g1 21?2g2 22?3g3 23?4g4 24?5g5 25?:31No