六自由度并联机器人基于Gramann-Cayley代数的奇异性条件
Patricia Ben-Horin和Moshe Shoham，会员，IEEE
摘要
本文研究了奇异性条件大多数的六自由度并联机器人在每一个腿上都有一个球形接头。首先,确定致动器螺丝在腿链中心。然后用凯莱代数和相关的分解方法用于确定哪些条件的导数(或刚度矩阵)包含这些螺丝是等级不足。这些工具是有利的,因为他们方便操纵坐标-简单的表达式表示的几何实体,从而使几何解释的奇异性条件是更容易获得。使用这些工具,奇异性条件(至少)144种这类的组合被划定在四个平面所相交的一个点上。这四个平面定义为这个零距螺丝球形关节的位置和方向。指数Terms-Gramann-Cayley代数，奇点，三条腿的机器。
一、介绍
在过去的二十年里,许多研究人员广泛研究并联机器人的奇异性。不像串联机器人,失去在奇异配置中的自由度,尽管并联机器人的执行器都是锁着但是他们的的自由度还是可以获得的。因此,这些不稳定姿势的全面知识为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。
主要的方法之一,用于寻找奇异性并行机器人是基于计算雅可比行列式进行的。Goelin和安杰利斯[1]分类奇异性的闭环机制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。当圣鲁克和Goelin[2]减少了算术操作要求定义的雅可比行列式高夫·斯图尔特平台(GSP),从而使数值计算得到多项式。
另一个重要的工具,为分析螺旋理论中的奇异性,首先阐述了1900的论文[6]和开发机器人应用程序。几项研究已经应用这个理论找到并联机器人的奇异性,例如,[11]-[14]。特别注意到情况,执行机构是线性和代表螺丝是零投的。在这些情况下,奇异的配置是解决通过使用几何,寻找可能的致动器线依赖[15]-[17]。其他分类方法闭环机制可以被发现在[18]-[22]。
在本文中,我们分析了奇异点的一大类三条腿的机器人,在每个腿链有一个球形接头上的任何点。我们只关注了正运动学奇异性。首先,我们发现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。然后我们使用Gramann-Cayley代数和相关的发展获得一个代数方程,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意义的奇异配置是最主要的一个优点,在这里将介绍其方法。
虽然之前的研究[53]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更广泛的一类机器人有三条腿和一个球形关节。使用降低行列式和Gramann-Cayley运营商我们获得一个通用的条件,这些机器人的奇异性提供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。第三节包含一个简短的在螺丝和大纲性质的背景下驱动器螺丝,零距螺丝作用于中心的球形关节。第四部分包含一个介绍Gramann-Cayley代数的基本工具用于寻找奇异性条件。这部分还包括刚度矩阵(或导数)分解成坐标自由表达。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自由度并联机器人有六间连通性基础和移动平台。肖海姆和罗斯[54]提供了调查可能的结构,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。GSP和三条腿的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了Podhorodeski和Pittens[55],他发现了一个类的三条腿的对称并联机器人,球形关节、转动关节的平台在每个腿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。
我们处理机器人有三个链连接到移动平台,每个驱动有两个1自由度关节或一个二自由度关节。这些链不一定是平等的,但都有移动和连接六个基地和之间的平台。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1自由度关节和最后两个二自由度的关节。所有的可能性都显示在表I和II。该列表只包含机器人,有平等的连锁,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,计算方式如下:
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惠到其他关节螺钉固定在链条。被动球形接头在每个链部队驱动器螺丝为零距(行)并且通过它的中心。因此,三个平面是创建中心位于自己的球形关节。
以下简要介绍了螺旋理论,广泛的解决[7],[73],[75];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,我们比较奇异条件在上述3 GSP平台与结果报告线几何方法。
根据相对几何条件的他行方法区分不同的几种类型沿着棱镜致动器[81]的奇异性。我们表明,所有这些奇异点是特定情况下的条件通过(17 c)提供,这是有效的三条腿以及6:3 GSP平台的机器人的考虑。这种结构的奇异的配置根据线几何分析包括五种类型:3 c、4 b、4 d,5 a和5 b[17],[36]。
四、奇异性分析
本节确定奇异性条件定义在第二节的机器人。第一部分包括寻找方向的执行机构的行动路线,基于解释第三节中介绍。他行通过球形接头中心,而他们的方向取决于关节的分布和位置。第二部分包括应用程序的方法使用了Gramann-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),所有例子的解决方案是象征性地平等,无论点位置的腿或腿的对称性。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
第一个例子是3-PRPS机器人提出Behi[61][见图3(a)]。对于每个腿驱动螺丝躺在这家由球形接头中心和转动关节轴。特别是,致动器螺杆是垂直于轴的,和致动器螺杆是垂直于轴的,这些方向被描绘在图3(b)。 第二个例子是the3-USR机器人提出Simaan et al。[66][见图4(a)]。每条腿有驱动器螺丝躺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3-PPSP Byun建造的机器人和[65][见图5(一个)]。每条腿,驱动螺丝躺在飞机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)。
图3 (a)3-PRPS机器人提出Behi[61]
(b)飞机和致动器螺丝
图4 (a)3自由度机器人提出Simaan和Shoham[66]
(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[65]
(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使用麦克米兰的分解,即(16)。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。拓扑,这个描述等于行6:3 GSP(或在[53]),这三条线,每经过一个双球面上的接头平台(见图6)。这意味着每对线共享一个公共点(这些点在图6中)。因此类的机器人被认为是在本文中,我们可以使用相同的标记点的至于6:3 GSP。六线与相关各机器人通过双点,并且,用同样的方式在图6。
图6 6 - 3 GSP
五、结果
本文提出一个广义奇异性分析并联机器人组成元素。这些是有一个球形接头在每个腿链的三条腿的6自由度机器人。因为球形关节需要驱动器，螺丝是纯粹的力量作用于他们的中心,他们的位置沿链是不重要的。组成元素包括144机制不同类型的关节,每个都有不同的联合装置沿链。提出并建立描述几个机器人出现在列表中。大量的机器人相关的分析组合不同被认为是。奇点的分析是由第一个找到的执行机构使用互惠的螺丝。然后,借助组合方法和Gramann-Cayley方法,得到刚度矩阵行列式在一个可以操作的协调自由形式,可以翻译成一个简单的几何条件之后。其定义是几何条件由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所有组成元素中的机器人。一个比较的结果与结果的奇点证明了其他技术所有先前描述奇异条件实际上是特殊情况下的几何条件的四架飞机交叉在一个点,一个条件获取的方法直接在这里提出。
Singularity Condition of Six-Degree-of-Freedom Three-Legged Parallel Robots Based on Gramann–Cayley Algebra Patricia Ben-Horin and Moshe Shoham, Aociate Member, IEEE
ABSTRACT This paper addrees the singularity condition of a broad cla 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 Gramann–Cayley algebra and the aociated 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 expreions 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 cla is delineated to be the intersection of four planes at one point.These four planes are defined by the locations of the spherical joints and the directions of the zero-pitch screws.Index Terms—Gramann–Cayley algebra, singularity, three-legged robots.
I.INTRODUCTION During 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 eential 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.Goelin and Angeles [1] claified the singularities of closed-loop mechanisms by considering two Jacobians that define the relationship between input and output velocities.St-Onge and Goelin [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 claification proposed by Goelin and Angeles to define six types of singularity that are derived using equations containing not only the input and output velocities but also explicit paive 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 are zero-pitched.In these cases, the singular configurations were solved by using line geometry, looking for poible actuator-line dependencies [15]–[17].Other approaches taken to claify singularities of closed-loop mechanisms can be found in [18]–[22].In this paper, we analyze the singularities of a broad cla 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 aociated 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 paing through the spherical joints.Then we use Gramann–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 cla of robots that have three legs with a spherical joints somewhere along the legs.Using the reduced determinant and Gramann–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 cla 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 Gramann–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-free expreions.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 poible structures that yield 6-DOF based on the mobility formula of Grübler and Kutzbach.They searched for all the poibilities 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 cla 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 discued 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 paive 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 claification, 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 necearily 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 poibilities are shown in Tables I and II.The list contains only the robots that have equal chains, totaling 144 different structures, but robots with any poible combination of chains can also be considered as membersof this cla.The total number of combinations, , is larger than 500 000, calculated as follows:
III.GOVERNING LINES This 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 paive spherical joint in each chain forces the actuator screws to have zero-pitch (lines) and to pa 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 addre the reciprocal screw systems of all the joints listed in Section II.
The geometric result for the singularity of the aforementioned cla 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 provided by (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 ANALYSIS This section determines the singularity condition for the cla 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 pa 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 Gramann–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, regardle 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 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 paing through the spherical joint center and containing the revolute joint axis.The actuator screw paes through the spherical joint center and intersects the revolute joint axis and.Similarly, the actuator screw paes 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 paing 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) 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 superbracket) 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 cla considered in this paper have two zero-pitch actuator screws paing through the spherical joint of each chain.Topologically, this description is equivalent to the lines of the 6-3 GSP (or in [53]), which has three pairs of lines, each paing 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 cla of robots considered in this paper, we can use the same notation of points as for the 6-3 GSP.The six lines aociated with each robot pa 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.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 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, with the aid of combinatorial methods and Gramann–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
