外文翻譯--六自由度機(jī)器人

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1、精選優(yōu)質(zhì)文檔-----傾情為你奉上 六自由度并聯(lián)機(jī)器人基于Grassmann-Cayley代數(shù)的奇異性條件 Patricia Ben-Horin和Moshe Shoham，會(huì)員，IEEE 摘要 本文研究了奇異性條件大多數(shù)的六自由度并聯(lián)機(jī)器人在每一個(gè)腿上都有一個(gè)球形接頭。首先,確定致動(dòng)器螺絲在腿鏈中心。然后用凱萊代數(shù)和相關(guān)的分解方法用于確定哪些條件的導(dǎo)數(shù)(或剛度矩陣)包含這些螺絲是等級(jí)不足。這些工具是有利的,因?yàn)樗麄兎奖悴倏v坐標(biāo)-簡(jiǎn)單的表達(dá)式表示的幾何實(shí)體,從而使幾何解釋的奇異性條件是更容易獲得。使用這些工具,奇異性條件(至少)144種這類的組合被劃定在四個(gè)平面所相交的一個(gè)點(diǎn)上。這四

2、個(gè)平面定義為這個(gè)零距螺絲球形關(guān)節(jié)的位置和方向。指數(shù)Terms-Grassmann-Cayley代數(shù)，奇點(diǎn)，三條腿的機(jī)器。 一、 介紹 在過(guò)去的二十年里,許多研究人員廣泛研究并聯(lián)機(jī)器人的奇異性。不像串聯(lián)機(jī)器人,失去在奇異配置中的自由度,盡管并聯(lián)機(jī)器人的執(zhí)行器都是鎖著但是他們的的自由度還是可以獲得的。因此,這些不穩(wěn)定姿勢(shì)的全面知識(shí)為提高機(jī)器人的設(shè)計(jì)和確定機(jī)器人的路徑規(guī)劃是至關(guān)重要的。 主要的方法之一,用于尋找奇異性并行機(jī)器人是基于計(jì)算雅可比行列式進(jìn)行的。Gosselin和安杰利斯[1]分類奇異性的閉環(huán)機(jī)制通過(guò)考慮兩個(gè)雅克比定義輸入速度和輸出速度之間的關(guān)系。當(dāng)圣魯克和Gosselin[2]

3、減少了算術(shù)操作要求定義的雅可比行列式高夫·斯圖爾特平臺(tái)(GSP),從而使數(shù)值計(jì)算得到多項(xiàng)式。 另一個(gè)重要的工具,為分析螺旋理論中的奇異性,首先闡述了1900的論文[6]和開發(fā)機(jī)器人應(yīng)用程序。幾項(xiàng)研究已經(jīng)應(yīng)用這個(gè)理論找到并聯(lián)機(jī)器人的奇異性,例如,[11]-[14]。特別注意到情況,執(zhí)行機(jī)構(gòu)是線性和代表螺絲是零投的。在這些情況下,奇異的配置是解決通過(guò)使用幾何,尋找可能的致動(dòng)器線依賴[15]-[17]。其他分類方法閉環(huán)機(jī)制可以被發(fā)現(xiàn)在[18]-[22]。 在本文中,我們分析了奇異點(diǎn)的一大類三條腿的機(jī)器人,在每個(gè)腿鏈有一個(gè)球形接頭上的任何點(diǎn)。我們只關(guān)注了正運(yùn)動(dòng)學(xué)奇異性。首先,我們發(fā)現(xiàn)螺絲相關(guān)執(zhí)行機(jī)

4、構(gòu)的每個(gè)鏈。因?yàn)槊恳粋€(gè)鏈包含一個(gè)球形接頭,自致動(dòng)器螺絲是相互聯(lián)合的,他們是通過(guò)球形關(guān)節(jié)的零螺距螺桿螺絲。然后我們使用Grassmann-Cayley代數(shù)和相關(guān)的發(fā)展獲得一個(gè)代數(shù)方程,它源于管理行機(jī)器人包含的剛度矩陣。直接和高效檢索的幾何意義的奇異配置是最主要的一個(gè)優(yōu)點(diǎn),在這里將介紹其方法。 雖然之前的研究[53]分析7架構(gòu)普惠制,各有至少三條并發(fā)關(guān)節(jié),本文擴(kuò)展了奇點(diǎn)分析程度更廣泛的一類機(jī)器人有三條腿和一個(gè)球形關(guān)節(jié)。使用降低行列式和Grassmann-Cayley運(yùn)營(yíng)商我們獲得一個(gè)通用的條件,這些機(jī)器人的奇異性提供在一個(gè)簡(jiǎn)單的幾何意義方式計(jì)算中。 本文的結(jié)構(gòu)如下。第二節(jié)詳細(xì)描述了運(yùn)動(dòng)學(xué)結(jié)構(gòu)的

5、并聯(lián)機(jī)器人。第三節(jié)包含一個(gè)簡(jiǎn)短的在螺絲和大綱性質(zhì)的背景下驅(qū)動(dòng)器螺絲,零距螺絲作用于中心的球形關(guān)節(jié)。第四部分包含一個(gè)介紹Grassmann-Cayley代數(shù)的基本工具用于尋找奇異性條件。這部分還包括剛度矩陣(或?qū)?shù))分解成坐標(biāo)自由表達(dá)。第五節(jié)中一個(gè)常見的例子給出了這種方法。最后,第六章比較了使用本方法結(jié)果與結(jié)果的其他技術(shù)。 二、 運(yùn)動(dòng)構(gòu)架 本文闡述了6自由度并聯(lián)機(jī)器人有六間連通性基礎(chǔ)和移動(dòng)平臺(tái)。肖海姆和羅斯[54]提供了調(diào)查可能的結(jié)構(gòu),產(chǎn)生基于流動(dòng)公式6自由度的Grubler和Kutzbach。他們尋找了所有的可能性,滿足這個(gè)公式對(duì)關(guān)節(jié)的數(shù)目和任何鏈接。GSP和三條腿的機(jī)器人結(jié)構(gòu)的一個(gè)子

6、集所列出的6自由度Shoham和羅斯。一個(gè)類似的例子也證實(shí)了了Podhorodeski和Pittens[55],他發(fā)現(xiàn)了一個(gè)類的三條腿的對(duì)稱并聯(lián)機(jī)器人,球形關(guān)節(jié)、轉(zhuǎn)動(dòng)關(guān)節(jié)的平臺(tái)在每個(gè)腿比其他結(jié)構(gòu)潛在有利。正如上面所討論的,大多數(shù)的報(bào)告文獻(xiàn)限制他們的分析結(jié)構(gòu)和球形關(guān)節(jié)位于移動(dòng)平臺(tái)和棱柱關(guān)節(jié)作為驅(qū)動(dòng)的關(guān)節(jié)。在這個(gè)分類,我們包括五種類型的關(guān)節(jié)和更多的可選職位的球形關(guān)節(jié)。 我們處理機(jī)器人有三個(gè)鏈連接到移動(dòng)平臺(tái),每個(gè)驅(qū)動(dòng)有兩個(gè)1自由度關(guān)節(jié)或一個(gè)二自由度關(guān)節(jié)。這些鏈不一定是平等的,但都有移動(dòng)和連接六個(gè)基地和之間的平臺(tái)。除了球形接頭(S),關(guān)節(jié)考慮是棱鏡(P),轉(zhuǎn)動(dòng)(R)、螺旋(H)、圓柱(C)和通用(U

7、),前三個(gè)是1自由度關(guān)節(jié)和最后兩個(gè)二自由度的關(guān)節(jié)。所有的可能性都顯示在表I和II。該列表只包含機(jī)器人,有平等的連鎖,總計(jì)144種不同的結(jié)構(gòu),但是機(jī)器人與任何可能的組合鏈也可以被認(rèn)為是membersof這類方法。組合的總數(shù),大于500 000,計(jì)算方式如下: 三、 管理方法 本節(jié)涉及螺絲和平臺(tái)運(yùn)動(dòng)的確定。因?yàn)榭紤]機(jī)器人有三個(gè)串行鏈,每個(gè)驅(qū)動(dòng)器螺絲的方向可以由其互惠到其他關(guān)節(jié)螺釘固定在鏈條。被動(dòng)球形接頭在每個(gè)鏈部隊(duì)驅(qū)動(dòng)器螺絲為零距(行)并且通過(guò)它的中心。因此,三個(gè)平面是創(chuàng)建中心位于自己的球形關(guān)節(jié)。 以下簡(jiǎn)要介紹了螺旋理論,廣泛的解決[7],[73],[75];我們解決在第二節(jié)中列出相互的

8、所有關(guān)節(jié)螺釘系統(tǒng)。 上述類的機(jī)器人的幾何結(jié)果奇點(diǎn)現(xiàn)在相比其他方法獲得的結(jié)果要準(zhǔn)確。首先,我們比較奇異條件在上述3 GSP平臺(tái)與結(jié)果報(bào)告線幾何方法。 根據(jù)相對(duì)幾何條件的他行方法區(qū)分不同的幾種類型沿著棱鏡致動(dòng)器[81]的奇異性。我們表明,所有這些奇異點(diǎn)是特定情況下的條件通過(guò)(17 c)提供,這是有效的三條腿以及6:3 GSP平臺(tái)的機(jī)器人的考慮。這種結(jié)構(gòu)的奇異的配置根據(jù)線幾何分析包括五種類型:3 c、4 b、4 d,5 a和5 b[17],[36]。 四、 奇異性分析 本節(jié)確定奇異性條件定義在第二節(jié)的機(jī)器人。第一部分包括尋找方向的執(zhí)行機(jī)構(gòu)的行動(dòng)路線,基于解釋第三節(jié)中介紹。

9、他行通過(guò)球形接頭中心,而他們的方向取決于關(guān)節(jié)的分布和位置。第二部分包括應(yīng)用程序的方法使用了Grassmann-Cayley代數(shù)在第四節(jié)定義奇點(diǎn)。因?yàn)槊繉?duì)線滿足在一個(gè)點(diǎn)(球形接頭),所有例子的解決方案是象征性地平等,無(wú)論點(diǎn)位置的腿或腿的對(duì)稱性。我們從文獻(xiàn)中舉例說(shuō)明使用三個(gè)機(jī)器人的解決方案。 1.方向的致動(dòng)器螺絲 第一個(gè)例子是3-PRPS機(jī)器人提出Behi[61][見圖3(a)]。對(duì)于每個(gè)腿驅(qū)動(dòng)螺絲躺在這家由球形接頭中心和轉(zhuǎn)動(dòng)關(guān)節(jié)軸。特別是,致動(dòng)器螺桿是垂直于軸的,和致動(dòng)器螺桿是垂直于軸的,這些方向被描繪在圖3(b)。 第二個(gè)例子是the3-USR機(jī)器人提出Simaan et al。[66]

10、[見圖4(a)]。每條腿有驅(qū)動(dòng)器螺絲躺在通過(guò)球形接頭中心和包含轉(zhuǎn)動(dòng)關(guān)節(jié)軸中。驅(qū)動(dòng)器螺絲穿過(guò)球形接頭中心并與轉(zhuǎn)動(dòng)關(guān)節(jié)軸相連。這些方向被描繪在圖4(b)。 第三個(gè)例子是3-PPSP Byun建造的機(jī)器人和[65][見圖5(一個(gè))]。每條腿,驅(qū)動(dòng)螺絲躺在飛機(jī)通過(guò)球形接頭中心和正常的棱鏡接頭軸。驅(qū)動(dòng)器螺絲垂直于軸的,和致動(dòng)器螺桿是垂直于軸的,這些方向被描繪在圖5(b)。 圖3 (a)3-PRPS機(jī)器人提出Behi[61] (b)飛機(jī)和致動(dòng)器螺絲 圖4 (a)3自由度機(jī)器人提出Simaan和Shoham[66] (b)飛機(jī)和致動(dòng)器螺絲的3自由度機(jī)器人 圖5 (a)

11、3-PPSP機(jī)器人提出Byun[65] (b)飛機(jī)和致動(dòng)器螺絲 2、 .奇異性條件 雅克(或superbracket)的機(jī)器人是分解成普通支架monomials使用麥克米蘭的分解,即(16)。解釋部分3—b機(jī)器人，本文認(rèn)為每個(gè)鏈有兩個(gè)零距驅(qū)動(dòng)器螺絲通過(guò)球形接頭。拓?fù)?這個(gè)描述等于行6:3 GSP(或在[53]),這三條線,每經(jīng)過(guò)一個(gè)雙球面上的接頭平臺(tái)(見圖6)。這意味著每對(duì)線共享一個(gè)公共點(diǎn)(這些點(diǎn)在圖6中)。因此類的機(jī)器人被認(rèn)為是在本文中,我們可以使用相同的標(biāo)記點(diǎn)的至于6:3 GSP。六線與相關(guān)各機(jī)器人通過(guò)雙點(diǎn),并且,用同樣的方式在圖6。 圖6 6 - 3 GSP

12、 五、 結(jié)果 本文提出一個(gè)廣義奇異性分析并聯(lián)機(jī)器人組成元素。這些是有一個(gè)球形接頭在每個(gè)腿鏈的三條腿的6自由度機(jī)器人。因?yàn)榍蛐侮P(guān)節(jié)需要驅(qū)動(dòng)器，螺絲是純粹的力量作用于他們的中心,他們的位置沿鏈?zhǔn)遣恢匾?。組成元素包括144機(jī)制不同類型的關(guān)節(jié),每個(gè)都有不同的聯(lián)合裝置沿鏈。提出并建立描述幾個(gè)機(jī)器人出現(xiàn)在列表中。大量的機(jī)器人相關(guān)的分析組合不同被認(rèn)為是。奇點(diǎn)的分析是由第一個(gè)找到的執(zhí)行機(jī)構(gòu)使用互惠的螺絲。然后,借助組合方法和Grassmann-Cayley方法,得到剛度矩陣行列式在一個(gè)可以操作的協(xié)調(diào)自由形式,可以翻譯成一個(gè)簡(jiǎn)單的幾何條件之后。其定義是幾何條件由執(zhí)行機(jī)構(gòu)位

13、置的線條和球形接頭,至少有一個(gè)相交點(diǎn)。這個(gè)有效的奇異點(diǎn)條件考慮所有組成元素中的機(jī)器人。一個(gè)比較的結(jié)果與結(jié)果的奇點(diǎn)證明了其他技術(shù)所有先前描述奇異條件實(shí)際上是特殊情況下的幾何條件的四架飛機(jī)交叉在一個(gè)點(diǎn),一個(gè)條件獲取的方法直接在這里提出。 Singularity Condition of Six-Degree-of-Freedom Three-Legged Parallel Robots Based on Grassmann–Cayley Algebra Patricia Ben-

14、Horin and Moshe Shoham, Associate Member, IEEE ABSTRACT This paper addresses the singularity condition of a broad class of six-degree-of-freedom three-legged parallel robots that have one spherical joint somewhere along each leg. First, the actuator screws for each leg-chain are determined. Then

15、 Grassmann–Cayley algebra and the associated superbracket decomposition are used to find the condition for which the Jacobian (or rigidity matrix) containing these screws is rank-deficient. These tools are advantageous since they facilitate manipulation of coordinate-free expressions representing ge

16、ometric entities, thus enabling the geometrical interpretation of the singularity condition to be obtained more easily. Using these tools, the singularity condition of (at least) 144 combinations of this class is delineated to be the intersection of four planes at one point. These four planes are de

17、fined by the locations of the spherical joints and the directions of the zero-pitch screws. Index Terms—Grassmann–Cayley algebra, singularity, three-legged robots. I. INTRODUCTION During the last two decades, many researchers have extensively investigated singularities of parallel robots. U

18、nlike serial robots that lose degrees of freedom (DOFs) in singular configurations, parallel robots might also gain DOFs even though their actuators are locked. Therefore, thorough knowledge of these unstable poses is essential for improving robot design and determining robot path planning. One of

19、the principal methods used for finding the singularities of parallel robots is based on calculation of the Jacobian determinant degeneracy. Gosselin and Angeles [1] classified the singularities of closed-loop mechanisms by considering two Jacobians that define the relationship between input and outp

20、ut velocities. St-Onge and Gosselin [2] reduced the arithmetical operations required to define the Jacobian determinant for the Gough–Stewart platform (GSP), and thus enabled numerical calculation of the obtained polynomial in real-time. Zlatanov et al. [3]–[5] expanded the classification proposed b

21、y Gosselin and Angeles to define six types of singularity that are derived using equations containing not only the input and output velocities but also explicit passive joint velocities. Another important tool that has served in the analysis of singularities is the screw theory, first expounde

22、d in Ball’s 1900 treatise [6] and developed for robotic applications by Hunt [7]–[9] and Sugimoto et al. [10]. Several studies have applied this theory to find singularities of parallel robots, for example, [11]–[14]. Special attention was paid to cases in which the actuators are linear and the repr

23、esenting screws are zero-pitched. In these cases, the singular configurations were solved by using line geometry, looking for possible actuator-line dependencies [15]–[17]. Other approaches taken to classify singularities of closed-loop mechanisms can be found in [18]–[22]. In this paper, we analyz

24、e the singularities of a broad class of three-legged robots, having a spherical joint at any point in each individual leg-chain. We focus only on forward kinematics singularities. First, we find the screws associated with the actuators of each chain. Since every chain contains a spherical joint, and

25、 since the actuator screws are reciprocal to the joint screws, they are zero-pitch screws passing through the spherical joints. Then we use Grassmann–Cayley algebra and related developments to get an algebraic equation which originates from the rigidity matrix containing the governing lines of the r

26、obot. The direct and efficient retrieval of the geometric meaning of the singular configurations is one of the main advantages of the method presented here. While the previous study [53] analyzed only seven architectures of GSP, each having at least three pairs of concurrent joints, this paper expa

27、nds the singularity analysis to a considerably broader class of robots that have three legs with a spherical joints somewhere along the legs. Using the reduced determinant and Grassmann–Cayley operators we obtain one single generic condition for which these robots are singular and provide in a simpl

28、e manner the geometric meaning of this condition. The structure of this paper is as follows. Section II describes in detail the kinematic architecture of the class of parallel robots under consideration. Section III contains a brief background on screws and outlines the nature of the actuator screw

29、s, which are zero-pitch screws acting on the centers of the spherical joints. Section IV contains an introduction to Grassmann–Cayley algebra which is the basic tool used for finding the singularity condition. This section also includes the rigidity matrix (or Jacobian) decomposition into coordinate

30、-free expressions. In Section V a general example of this approach is given. Finally, Section VI compares the results obtained using the present method with results obtained by other techniques. II. KINEMATIC ARCHITECTURE This paper deals with 6-DOF parallel robots that have connectivity six bet

31、ween the base and the moving platform. Shoham and Roth [54] provided a survey of the possible structures that yield 6-DOF based on the mobility formula of Grübler and Kutzbach. They searched for all the possibilities that satisfy this formula with respect to the number of joints connected to any of

32、the links. The GSP and three-legged robots are a subset of the structures with 6-DOF listed by Shoham and Roth. A similar enumeration was provided also by Podhorodeski and Pittens [55], who found a class of three-legged symmetric parallel robots that have spherical joints at the platform and revolut

33、e joints in each leg to be potentially advantageous over other structures. As discussed above, most of the reports in the literature limit their analysis to structures with spherical joints located on the moving platform and revolute or prismatic joints as actuated or passive additional joints. Exce

34、ptions are the family of 14 robots proposed by Simaan and Shoham [28] which contain spherical-revolute dyads connected to the platform, and some structures mentioned below which have revolute or prismatic joints on the platform. In this classification, we include five types of joints and more option

35、al positions for the spherical joints. We deal with robots that have three chains connected to the moving platform, each actuated by two 1-DOF joints or one 2-DOF joint. These chains are not necessarily equal, but all have mobility and connectivity six between the base and the platform. Besides the

36、 spherical joint (S), the joints taken into consideration are prismatic (P), revolute (R), helical (H), cylindrical (C), and universal (U), the first three being 1-DOF joints and the last two being 2-DOF joints. All the possibilities are shown in Tables I and II. The list contains only the robots th

37、at have equal chains, totaling 144 different structures, but robots with any possible combination of chains can also be considered as membersof this class. The total number of combinations, , is larger than 500 000, calculated as follows: III. GOVERNING LINES This section deals with the screws t

38、hat determine the platform motion. Since the robots under consideration have three serial chains, the direction of each actuator screw can be determined by its reciprocity to the other joint screws in the chain. The passive spherical joint in each chain forces the actuator screws to have zero-pitch

39、(lines) and to pass through its center. Therefore, three flat pencils are created having their centers located at the spherical joints. Following a brief introduction to the screw theory that is extensively treated in [7], [73]–[75]; we address the reciprocal screw systems of all the joints lis

40、ted in Section II. The geometric result for the singularity of the aforementioned class of robots is now compared with the results obtained by other approaches in the literature. First, we compare the singularity condition described above for the 6-3 GSP platform with the results reported for t

41、he line geometry method. The line geometry method distinguishes among several types of singularities, according to the relative geometric condition of he lines along the prismatic actuators [81]. We show that all these singularities are particular cases of the condition provided by (17c), whic

42、h is valid for the three-legged robots under consideration as well as for the 6-3 GSP platform. The singular configurations of this structure according to line geometry analysis include five types: 3C, 4B, 4D, 5A, and 5B [17], [36]. IV. SINGULARITY ANALYSIS This section determines the singularit

43、y condition for the class of robots defined in Section II. The first part consists of finding the direction of the actuator lines of action, based on the explanation introduced in Section III. The lines pass through the spherical joint center while their directions depend on the distribution and pos

44、ition of the joints. The second part includes application of the approach using Grassmann–Cayley algebra presented in Section IV for defining singularity when considering six lines attaching two platforms. Since every pair of lines meet at one point (the spherical joint), the solution for all the ca

45、ses is symbolically equal, regardless of the points’ location in the leg or the symmetry of the legs. We exemplify the solution using three robots from the literature. A. Direction of the Actuator Screws The first example is the 3-PRPS robot as proposed by Behi [61] [see Fig. 3(a)]. For each leg t

46、he actuated screws lie on theplane defined by the spherical joint center and the revolute joint axis. In particular,the actuator screw is perpendicular to the axis of , and the actuator screw is perpendicular to the axis of , these directions being depicted in Fig. 3(b). The second example is the3-

47、USR robot as proposed by Simaan et al.[66][see Fig.4(a)].Every leg has the actuator screws lying on the plane passing through the spherical joint center and containing the revolute joint axis. The actuator screw passes through the spherical joint center and intersects the revolute joint axis and. Si

48、milarly, the actuator screw passes through the spherical joint center and intersects the revolute joint axis and , these directions being depicted in Fig. 4(b). The third example is the 3-PPSP robot built by Byun and Cho [65] [see Fig. 5(a)]. For every leg the actuated screws lie on the plane passi

49、ng through the spherical joint center and being normal to the prismatic joint axis.The actuator screw is perpendicular to the axis of , and the actuator screw is perpendicular to the axis of , these directions being depicted in Fig. 5(b). Fig. 3. (a) The 3-PRPS robot as proposed by Behi [61]. (b)

50、 Planes and actuator screws. Fig. 4. (a) The 3-USR robot as proposed by Simaan and Shoham [66]. (b) Planes and actuator screws of the 3-USR robot. Fig. 5. (a) 3-PPSP robot as proposed by Byun and Cho [65]. (b) Planes and actuator screws. B. Singularity Condition The Jacobian (or supe

51、rbracket) of a robot is decomposed into ordinary bracket monomials using McMillan’s decomposition, namely (16). As explained in Section III-B, all the robots of the class considered in this paper have two zero-pitch actuator screws passing through the spherical joint of each chain. Topologically, th

52、is description is equivalent to the lines of the 6-3 GSP (or in [53]), which has three pairs of lines, each passing through a double spherical joint on the platform (see Fig. 6). This means that each pair of lines share one common point (in Fig. 6 these points are , , and ). Therefore for the class

53、of robots considered in this paper, we can use the same notation of points as for the 6-3 GSP. The six lines associated with each robot pass through the pairs of points,and , in the same way as in Fig. 6. Due to the common points of the pairs of lines ,and ,denoted , and respectively, many of the mo

54、nomials of (16) vanish due to (4). Fig. 6. 6-3 GSP. V. CONCLUSION This paper presents singularity analysis for a broad family of parallel robots. These are 6-DOF three-legged robots which have one spherical joint in each leg-chain. Since the spherical joints entail the actuato

55、r screws to be pure forces acting on their centers, their location along the chain is not important. The family includes 144 mechanisms incorporating diverse types of joints that each has a different joint arrangement along the chains. Several proposed and built robots described in the literature ap

56、pear in this list. A larger number of robots are relevant to this analysis if combinations of different legs are considered. The singularity analysis was performed by first finding the lines of action of the actuators using the reciprocity of screws. Then, with the aid of combinatorial methods and G

57、rassmann–Cayley operators, the rigidity matrix determinant was obtained in a manipulable coordinate-free form that could be translated later into a simple geometric condition. The geometric condition consists of four planes, defined by the actuator lines and the position of the spherical joints, whi

58、ch intersect at least one point. This singularity condition is valid for all the robots in the family under consideration.A comparison of this singularity result with results obtained by other techniques demonstrated that all the previously described singularity conditions are actually special cases of the geometrical condition of four planes intersecting at a point, a condition that was obtained straightforwardly by the method suggested here 專心---專注---專業(yè)