外文翻譯--六自由度機器人

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1、精選優(yōu)質文檔-----傾情為你奉上 六自由度并聯(lián)機器人基于Grassmann-Cayley代數(shù)的奇異性條件 Patricia Ben-Horin和Moshe Shoham，會員，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)機器人在每一個腿上都有一個球形接頭。首先,確定致動器螺絲在腿鏈中心。然后用凱萊代數(shù)和相關的分解方法用于確定哪些條件的導數(shù)(或剛度矩陣)包含這些螺絲是等級不足。這些工具是有利的,因為他們方便操縱坐標-簡單的表達式表示的幾何實體,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平面所相交的一個點上。這四

2、個平面定義為這個零距螺絲球形關節(jié)的位置和方向。指數(shù)Terms-Grassmann-Cayley代數(shù)，奇點，三條腿的機器。 一、 介紹 在過去的二十年里,許多研究人員廣泛研究并聯(lián)機器人的奇異性。不像串聯(lián)機器人,失去在奇異配置中的自由度,盡管并聯(lián)機器人的執(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)定姿勢的全面知識為提高機器人的設計和確定機器人的路徑規(guī)劃是至關重要的。 主要的方法之一,用于尋找奇異性并行機器人是基于計算雅可比行列式進行的。Gosselin和安杰利斯[1]分類奇異性的閉環(huán)機制通過考慮兩個雅克比定義輸入速度和輸出速度之間的關系。當圣魯克和Gosselin[2]

3、減少了算術操作要求定義的雅可比行列式高夫·斯圖爾特平臺(GSP),從而使數(shù)值計算得到多項式。 另一個重要的工具,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)機器人應用程序。幾項研究已經(jīng)應用這個理論找到并聯(lián)機器人的奇異性,例如,[11]-[14]。特別注意到情況,執(zhí)行機構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使用幾何,尋找可能的致動器線依賴[15]-[17]。其他分類方法閉環(huán)機制可以被發(fā)現(xiàn)在[18]-[22]。 在本文中,我們分析了奇異點的一大類三條腿的機器人,在每個腿鏈有一個球形接頭上的任何點。我們只關注了正運動學奇異性。首先,我們發(fā)現(xiàn)螺絲相關執(zhí)行機

4、構的每個鏈。因為每一個鏈包含一個球形接頭,自致動器螺絲是相互聯(lián)合的,他們是通過球形關節(jié)的零螺距螺桿螺絲。然后我們使用Grassmann-Cayley代數(shù)和相關的發(fā)展獲得一個代數(shù)方程,它源于管理行機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優(yōu)點,在這里將介紹其方法。 雖然之前的研究[53]分析7架構普惠制,各有至少三條并發(fā)關節(jié),本文擴展了奇點分析程度更廣泛的一類機器人有三條腿和一個球形關節(jié)。使用降低行列式和Grassmann-Cayley運營商我們獲得一個通用的條件,這些機器人的奇異性提供在一個簡單的幾何意義方式計算中。 本文的結構如下。第二節(jié)詳細描述了運動學結構的

5、并聯(lián)機器人。第三節(jié)包含一個簡短的在螺絲和大綱性質的背景下驅動器螺絲,零距螺絲作用于中心的球形關節(jié)。第四部分包含一個介紹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(或導數(shù))分解成坐標自由表達。第五節(jié)中一個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果與結果的其他技術。 二、 運動構架 本文闡述了6自由度并聯(lián)機器人有六間連通性基礎和移動平臺。肖海姆和羅斯[54]提供了調查可能的結構,產生基于流動公式6自由度的Grubler和Kutzbach。他們尋找了所有的可能性,滿足這個公式對關節(jié)的數(shù)目和任何鏈接。GSP和三條腿的機器人結構的一個子

6、集所列出的6自由度Shoham和羅斯。一個類似的例子也證實了了Podhorodeski和Pittens[55],他發(fā)現(xiàn)了一個類的三條腿的對稱并聯(lián)機器人,球形關節(jié)、轉動關節(jié)的平臺在每個腿比其他結構潛在有利。正如上面所討論的,大多數(shù)的報告文獻限制他們的分析結構和球形關節(jié)位于移動平臺和棱柱關節(jié)作為驅動的關節(jié)。在這個分類,我們包括五種類型的關節(jié)和更多的可選職位的球形關節(jié)。 我們處理機器人有三個鏈連接到移動平臺,每個驅動有兩個1自由度關節(jié)或一個二自由度關節(jié)。這些鏈不一定是平等的,但都有移動和連接六個基地和之間的平臺。除了球形接頭(S),關節(jié)考慮是棱鏡(P),轉動(R)、螺旋(H)、圓柱(C)和通用(U

7、),前三個是1自由度關節(jié)和最后兩個二自由度的關節(jié)。所有的可能性都顯示在表I和II。該列表只包含機器人,有平等的連鎖,總計144種不同的結構,但是機器人與任何可能的組合鏈也可以被認為是membersof這類方法。組合的總數(shù),大于500 000,計算方式如下: 三、 管理方法 本節(jié)涉及螺絲和平臺運動的確定。因為考慮機器人有三個串行鏈,每個驅動器螺絲的方向可以由其互惠到其他關節(jié)螺釘固定在鏈條。被動球形接頭在每個鏈部隊驅動器螺絲為零距(行)并且通過它的中心。因此,三個平面是創(chuàng)建中心位于自己的球形關節(jié)。 以下簡要介紹了螺旋理論,廣泛的解決[7],[73],[75];我們解決在第二節(jié)中列出相互的

8、所有關節(jié)螺釘系統(tǒng)。 上述類的機器人的幾何結果奇點現(xiàn)在相比其他方法獲得的結果要準確。首先,我們比較奇異條件在上述3 GSP平臺與結果報告線幾何方法。 根據(jù)相對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器[81]的奇異性。我們表明,所有這些奇異點是特定情況下的條件通過(17 c)提供,這是有效的三條腿以及6:3 GSP平臺的機器人的考慮。這種結構的奇異的配置根據(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)的機器人。第一部分包括尋找方向的執(zhí)行機構的行動路線,基于解釋第三節(jié)中介紹。

9、他行通過球形接頭中心,而他們的方向取決于關節(jié)的分布和位置。第二部分包括應用程序的方法使用了Grassmann-Cayley代數(shù)在第四節(jié)定義奇點。因為每對線滿足在一個點(球形接頭),所有例子的解決方案是象征性地平等,無論點位置的腿或腿的對稱性。我們從文獻中舉例說明使用三個機器人的解決方案。 1.方向的致動器螺絲 第一個例子是3-PRPS機器人提出Behi[61][見圖3(a)]。對于每個腿驅動螺絲躺在這家由球形接頭中心和轉動關節(jié)軸。特別是,致動器螺桿是垂直于軸的,和致動器螺桿是垂直于軸的,這些方向被描繪在圖3(b)。 第二個例子是the3-USR機器人提出Simaan et al。[66]

10、[見圖4(a)]。每條腿有驅動器螺絲躺在通過球形接頭中心和包含轉動關節(jié)軸中。驅動器螺絲穿過球形接頭中心并與轉動關節(jié)軸相連。這些方向被描繪在圖4(b)。 第三個例子是3-PPSP Byun建造的機器人和[65][見圖5(一個)]。每條腿,驅動螺絲躺在飛機通過球形接頭中心和正常的棱鏡接頭軸。驅動器螺絲垂直于軸的,和致動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機器人提出Behi[61] (b)飛機和致動器螺絲 圖4 (a)3自由度機器人提出Simaan和Shoham[66] (b)飛機和致動器螺絲的3自由度機器人 圖5 (a)

11、3-PPSP機器人提出Byun[65] (b)飛機和致動器螺絲 2、 .奇異性條件 雅克(或superbracket)的機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機器人，本文認為每個鏈有兩個零距驅動器螺絲通過球形接頭。拓撲,這個描述等于行6:3 GSP(或在[53]),這三條線,每經(jīng)過一個雙球面上的接頭平臺(見圖6)。這意味著每對線共享一個公共點(這些點在圖6中)。因此類的機器人被認為是在本文中,我們可以使用相同的標記點的至于6:3 GSP。六線與相關各機器人通過雙點,并且,用同樣的方式在圖6。 圖6 6 - 3 GSP

12、 五、 結果 本文提出一個廣義奇異性分析并聯(lián)機器人組成元素。這些是有一個球形接頭在每個腿鏈的三條腿的6自由度機器人。因為球形關節(jié)需要驅動器，螺絲是純粹的力量作用于他們的中心,他們的位置沿鏈是不重要的。組成元素包括144機制不同類型的關節(jié),每個都有不同的聯(lián)合裝置沿鏈。提出并建立描述幾個機器人出現(xiàn)在列表中。大量的機器人相關的分析組合不同被認為是。奇點的分析是由第一個找到的執(zhí)行機構使用互惠的螺絲。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可以操作的協(xié)調自由形式,可以翻譯成一個簡單的幾何條件之后。其定義是幾何條件由執(zhí)行機構位

13、置的線條和球形接頭,至少有一個相交點。這個有效的奇異點條件考慮所有組成元素中的機器人。一個比較的結果與結果的奇點證明了其他技術所有先前描述奇異條件實際上是特殊情況下的幾何條件的四架飛機交叉在一個點,一個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)