基于Grassmann-Cayley代数的六自由度三足并联机器人的奇异性条件Patricia Ben-Horin，Moshe Shoham，IEEE准会员关键词：指数，Terms-Grassmann-Cayley代数，奇异性条件，六自由度三足机器人摘要本文研究了每一个腿上都有一个球形接头的大多数六自由度并联机器人的奇异性条件。
首先,应该确定致动器螺丝位于腿链中心，然后在使用基于Grassmann-Cayley代数和相关的分解方法来确定这些螺丝包含的哪些条件是导数刚度等级不足的。
这些工具是非常有用,因为他们可以方便的表示坐标-并用简单的表达式来表示几何实体,从而使用几何解释奇异性条件是更容易获得。
利用这些工具, 这类奇异性条件的144种组合被划定在四个平面所相交的一个点上。
这四个平面被定义为这个零距螺丝球形关节的位置和方向。
一、介绍在过去的二十年里,许多研究人员一直在广泛地研究并联机器人的奇异性。
不像串联机器人, 尽管并联机器人失去了在奇异配置中的自由度,而且执行器都是锁定的，但是他们的的自由度还是可以获得的。
因此,对这些不稳定姿势的机器人的全面研究对于提高机器人的设计和确定机器人的路径规划是至关重要的。
用于寻找并联机器人奇异性的主要的方法之一,是基于计算雅可比行列式进行计算。
Gosselin和安杰利斯的分类奇异性的闭环机制是通过考虑两个雅克比定义输入速度和输出速度之间的关系。
圣鲁克和Gosselin减少了定义的雅可比行列式算术操作要求,从而通过数值计算得到多项式。
另一个重要的工具,是用螺旋理论分析奇异性,在1900的论文中中开发机器人的相关应用程序，有几项研究已经应用这个理论找到并联机器人的奇异性。
在论文中，特别注意到情况是执行机构是线性和代表螺丝是零投的。
在这些情况下,奇异的配置是解决是使用几何方法寻找可能的致动器线依赖，可以发现其他分类方法闭环机制。
在本文中,我们分析了三足的机器人的一大类奇异点,机器人每个腿链有一个球形接头，我们只关注了其中正运动学奇异性。
首先,我们研究了螺丝相关执行机构的每个链，因为每一个链包含一个球形接头,自致动器螺丝是相互联合的,他们是通过球形关节的零螺距螺杆螺丝联合。
然后我们使用Grassmann-Cayley 代数和相关的发展理论获得一个代数方程,它源于管理行机器人包含的刚度矩阵。
直接和高效检索的几何奇异配置是这种方法最主要的一个优点,在这里将介绍其方法。
虽然之前的研究分析7架构普惠制,各有至少三条并发关节,本文扩展了奇点分析程度，分析了更广泛的一类具有三足和一个球形关节的机器人。
使用降低行列式和Grassmann-Cayley的运算方式我们获得了一个通用的条件,这些机器人的奇异性隐含在一个简单的几何意义方式计算中。
本文的结构如下。
第二节详细描述了运动学结构的并联机器人。
第三节包含了在一个简短的螺丝和大纲性质的背景下的驱动器螺丝、零距螺丝共同作用于中心的球形关节。
第四部分介绍用Grassmann-Cayley代数的基本工具来寻找奇异性条件。
这部分还包括刚度矩阵(或导数)分解成自由坐标表达。
第五节中一个常见的例子给出了这种方法。
最后,第六章比较了使用本方法结果与结果的其他技术。
二、运动构架本文阐述了6自由度并联机器人的六间连通性基础和移动平台。
肖海姆和罗斯提调查了可能的结构,产生基于流动公式6自由度的Grubler和Kutzbach。
他们寻找了所有的可能性,满足这个公式对关节的数目和任何链接。
包括GSP和三足机器人结构的一个子集所列出的6自由度Shoham和罗斯。
一个类似的例子也证实了Podhorodeski和Pittens,他发现了一个类的三足的对称并联机器人, 在每个腿上的球形关节、转动关节的平台都比其他结构潜在有利。
正如上面所讨论的,大多数的报告文献限制他们的分析结构和球形关节位于移动平台和棱柱关节作为驱动的关节。
在这个分类,我们机器人包括五种类型的关节和更多的可选职位的球形关节。
我们处理的机器人有三个链连接到移动平台,每个驱动有两个一自由度关节或一个二自由度关节。
这些链不一定是平等的,但都可以移动和连接六个基础和其之间的平台。
除了球形接头(S),关节考虑是棱镜(P),转动(R)、螺旋(H)、圆柱(C)和通用(U),前三个是一自由度关节，最后是两个二自由度的关节。
所有的可能性都显示在表I和II。
该列表只包含有平等连锁的机器人,总计144种不同的结构,但是机器人与任何可能的组合链也可以被认为是membersof这类方法。
组合的总数,大于500 000。
三、管理方法本节涉及螺丝和平台运动的确定。
因为考虑机器人有三个串行链,每个驱动器螺丝的方向可以由其互换到其他关节螺钉固定的链条。
被动球形接头在每个链组驱动器螺丝为零距(行)并且通过它的中心。
因此,三个平面是创建于自己中心的球形关节。
以下简要介绍了螺旋理论的广泛解决，用于我们解决在第二节中列出相互的所有关节螺钉系统。
上述类的机器人的几何结果奇点现在相比其他方法获得的结果要准确。
首先,我们比较奇异条件位于上述3个GSP平台与结果报告线几何方法。
用基于相对几何条件的方法区分不同类型的机器人沿着棱镜致动器的奇异性。
我们发现,所有这些奇异点是特定情况下的条件通过(17 c)提供,这是基于有效的三足以及6:3 GSP平台的机器人的考虑。
这种结构的奇异配置根据线几何分析得来的，包括五种类型:图-c、图4-b、图4 -d,图-5 a和图5-b。
四、奇异性分析本节确定的奇异性条件定义包括两部分。
第一部分包括寻找方向的执行机构的行动路线,是基于解释第三节中介绍的，它通过球形接头中心,而它们的方向取决于机器人关节的分布和位置。
第二部分创建应用程序的方法则是使用了Grassmann-Cayley代数在第四节定义奇点。
因为每对线满足在一个点(球形接头)的所有例子的解决方案是平等的,无论点位置或者腿是否具有对称性。
我们从文献中举例说明使用三个机器人的解决方案。
1.方向的致动器螺丝第一个例子[见图3(a)]。
每个腿驱动螺丝安装在球形接头中心和转动关节轴上。
特别注意的是是,致动器螺杆是垂直于轴的,这些方向被描绘在图3(b)。
第二个例子是Simaan etal提出的the3-USR机器人。
[见图4(a)]。
每条腿和驱动器螺丝安装在通过球形接头中心和包含转动关节轴中。
驱动器螺丝穿过球形接头中心并与转动关节轴相连。
这些方向被描绘在图4(b)。
第三个例子是Byun建造的3-PPSP机器人[见图5]。
每条腿和驱动螺丝的安装通过球形接头中心和正常的棱镜接头轴。
驱动器螺丝垂直于轴的,和致动器螺杆是垂直于轴的,这些方向被描绘在图5(b)。
3-(b)飞机和致动器螺丝4-(b)飞机和致动器螺丝的3自由度机器人图5-(a) 3-PPSP机器人提出Byun 5-(b)飞机和致动器螺丝2、.奇异性条件雅克(或superbracket)的机器人是分解成普通支架monomials并使用麦克米兰来分解的，用于解释部分3—b机器人，本文认为每个链有两个零距驱动器螺丝通过球形接头。
这三条线,每经过一个双球面上的接头平台(见图6，就这意味着每对线共享一个公共点(这些点在图6中)。
因此在本文中此类的机器人被认为是我们可以使用相同的标记点标记的6:3 GSP。
六线与相关各机器人通过双点,我们用同样的方式标记在图6。
图6 6 - 3 GSP五、结果本文提出了用一个广义奇异性分析研究并联机器人组成元素。
这些是每个腿链有一个球形接头的三足6自由度机器人。
因为球形关节需要驱动器，螺丝是通过纯粹的力作用于他们的中心,他们的位置沿链是不重要的。
组成元素包括144个机制不同类型的关节,每个关节都有不同的联合装置沿着运动链。
在列表提出并建立描述了几个机器模型。
大量机器人的相关分析组合发现认为是奇点的分析是取决于第一个找到的执行机构使用的相互作用的螺丝。
然后,借助组合方法和Grassmann-Cayley方法,得到刚度矩阵行列式和一个可以操作的协调自由形式,这可以理解为在获得一个简单的几何条件之后得出。
其定义的几何条件是，执行机构位置的线条和球形接头至少有一个相交点。
这个有效的奇异点条件考虑到所有组成元素中的机器人。
比较的结果与结果的奇点证明了其他所有先前技术描述的奇异条件实际上是特殊情况下的几何交叉点,而一个基于Grassmann-Cayley 代数的六自由度三足并联机器人的奇异性条件在这里被提出。
Singularity Condition ofSix-Degree-of-Freedom Three-Legged Parallel Robots BasedonGrassmann–Cayley Algebra Patricia Ben-Horin and Moshe Shoham, Associate Member, IEEEABSTRACTThis 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 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 geometric 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 aredefined 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. INTRODUCTIONDuring the last two decades, many researchers have extensively investigated singularities of parallel robots. Unlike 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 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 output 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 by 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 expounded 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 representing screws arezero-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 analyze 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 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 robot. 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 expands 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 simple 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 screws, 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-freeexpressions. 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 between 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 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 revolute 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. Exceptions 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 optional 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 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 Iand II. The list contains only the robots that 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 LINESThis section deals with the screws that 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 (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 listed 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 the 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 providedby (17c), which 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 ANALYSISThis section determines the singularity 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 position 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 cases 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 ScrewsThe first example is the 3-PRPS robot as proposed by Behi [61] [see Fig. 3(a)]. For each leg the 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-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. Similarly, 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 passing 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 inFig. 5(b).Fig. 3. (a) The 3-PRPS robot as proposed by Behi [61]. (b) Planes and actuatorscrews.actuator screws of the 3-USR robot.Fig. 5. (a) 3-PPSP robot as proposed by Byun and Cho [65]. (b) Planes and actuatorscrews.B. Singularity ConditionThe Jacobian (or superbracket) of a robot is decomposed into ordinary bracket monom ials 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. T opologically, this 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 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 monomials of (16) vanish due to (4).Fig. 6. 6-3 GSP.V. CONCLUSIONThis 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 actuator 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 appear 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, withthe aid of combinatorial methods and Grassmann–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, which 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。
