并联位移机器人的外文文献翻译、中英文翻译、外文翻译

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并联位移机器人的设计

Jacques M.HERVE

ECELE CENTRALE PARIS

92295 CHATENAY MALABRY CEDEX

FRANCE

摘要:本文目的是对偶具有人性化机器人的应用做一个完全的介绍,并将着重讨论并行机器人特别是那些能够进行空间平移的机器人。在许多工业的应用过程中这种机器人被证明其末端执行器在空间上的定位是没必要的。这个方法的优点是我们能系统地导出能预期得到位移子群的所有运动学链。因此，我们调查了机器人的整个家族。T-STAR机器人现在就是一台工作装置。而H-ROBOT,PRISM-ROBOT是新的可能的机器人。这些机器人能满足现代生产快节奏工作中价格低以及符合挑选的工作环境，如选料、安排、包装、装配等发日益增长的需求。

关键词：运动学,并行机器人

引言

群论可以运用于一系列位移当中。根据这个理论，如果我们能够证明群{D}包含所有的可能的位移，那么{D}就具有群结构。刚体的最显著运动是由群{D}表现出来的。这方法导致机械装置的分类 [1]。建立这样的一个分类的主要的步骤是将位移群的所有子群导出。这能通过检验所有具有旋转和平移特性的[2]产品直接推理出。然而,一个更有效的方法存在于假设群论[3],[4]中。假设群论是在取决于许多有限实参数的全纯映射的基础上定义的。位移群{D}是六维假设群的一个特例。

假设理论

在假设群论的框架内,我们将用于补偿李代数的微元变换与通过其前面幂运算得到的有限运算结合起来。连续群通过与群微元变换有关的微分幂运算描述出来。

另外，群体特性通过微分运算及其逆运算所得到的李代数的代数结构而得到了解释。让我们回忆一下李代数主要的定义公理：一个李代数是一个具有封闭乘积的反对偶称双线性的矢量空间。众所周知 [5]，螺旋速度场是在给定点N的条件下通过运算得到的一个六维的矢量空间。由下面［3］中步骤表明，我们能得完整的欧几里得位移{D}子群列表（见大纲表1）。该列表是通过首先定义一个与速度场有关的微分运算符得到的。然后，通过幂运算，得到了李代数有限位移的表达式。此表达式相当于仿射的直接归一正交变换。螺旋速度场的子李代数是对偶位移子群组的直接描述。

{X (w)}子群

为了利用平行机理得到空间平移，我们需要找到所有位移子群的交集——空间平移子群{T}。我们考虑的子群交集将严格的包含于两个“平行”子群内。此类别的最重要的情况是2个{X (w)} 子群和2个不同矢量方向w和w’的平行关系。这很容易证明：

{X(w)} {X(w’)}={T},w≠w’

子群{X (w)}在机制设计起一个很重要的作用。该子群由带有旋转运动的空间平移组成,其旋转主轴方向与所给定的矢量w的方向始终平行。{X(w)}机械联系的实际实施是通过子群{X(w)}代表的系列运动学对偶中的命令实现的。实际上棱柱对偶和旋转对偶P,R,H都用于构造机器人(圆柱体对偶C以紧凑的方式结合棱柱对偶和旋转对偶)。产生的这些运动学对偶的所有可能组合由子群组{X (w)}在[6]中给出。

同时它们必须连续的满足两种几何情况：旋转轴与螺旋轴要与给定的矢量w平行；不是被动运动。

{X{w}}子群的位移运算符,在M点的作用是:

M → N + ａu + bv + cw +exp(hw^) N M

^是矢量乘积标志。

点N和矢量u,v,w组成了空间的正交标架的基准。a, b, c, h为具有四维空间的子群的四个参数。

空间平移的并联机器人

当两子群组{X(w)} 和{X(w’)},w≠w’,满足w≠w’，但矢量平行时，在移动平台和固定马达之间，其机械生成元就足以能产生空间平移。三个子群组{X

(w)},{X(w’)},{X(w’’)},w≠w’时其生成元同样也能产生空间平移。P，R或H的任何系列组成群组{X (w)}生成元的对偶的空间平移都能被实现。此外，这3种机械生成元可以是不同或一样但都取决于所需的运动学结果。这种组合范围很广，使得整个能进行空间平移的机器人家族成员得到了增加。最有趣的是建筑的模拟能容易地是完成，机器手的选择也能适应委员的需要。Clavel的Delta机器人属于这个家族，因为它基于相同的运动学原理[7]。

并行操作机器人Y-STAR

STAR [16] 由3个能产生{X (u)}, {X (u’)}, {X(u’’)} (fig 1)子群组的协作操作臂组成。3只机械臂是相同且每只都能通过一系列的RHPaR生成一个子群{X (u)}，其中Pa代表循环平移协作，此平移协作由一块绞接的平行四边形的两对偶立的杆控制决定。

两旋转对偶轴与螺旋对偶轴必须平行以保证能生成{X (u)}子群组。每条机械臂，第一个2对偶,即同轴旋转对偶和螺旋对偶组成固定机器人的固定部分，同时形成处于相同平面的轴的机械结构，将其分为三个相同部分，从而形成了Y行状。因此任意两轴之间的角度都占整个空间角度的2 /3。机器人的移动部分由PaR系列组成，都能集中于移动平台做指定的某点位置。平台与参考平面保持平行，不能绕垂直于参考平面的轴旋转。任何的一种专有的末端执行器都能是放置在这流动的平台上。所得到的反应移动平台的{T}子群仅能在空间进行平移,在[8]中给出。

H型机器人

大部分并型机器人包括Delta机器人和Y Star机器人，其末端执行器的工作空间与整个装置相比较小。这是此类机器人的一个缺陷。为了避免这种工作空间的限制，对偶此装置安装具有平行轴的电动千斤顶。与Y Star相似的机器人臂不能使用：三个相同集{X (v)}的交集等于{X (v)}而不是{T}。因此，在计新的H机器人[16]时，我们选择与Y-Sta相同的两条手臂，第三条手臂可与Delta手臂相比。这第三条机械臂开始形成带有与第一个两电动

千斤顶平行的机动化柱状对偶的固定框架。继以之绞接的二维平行四边形，此四边形由于其中一根杆的缘故能绕垂直于P对偶的轴转动。与此杆相对偶的杆经由平行轴的旋转对偶R

被连结到移动平台上。当平行四边形形状变化时，这个性质被保持(自由度为一)。此机器人的第一个样机有一个团队的学生在Pastoré教授的指导下于法国“IUT de V ille D’Avray”完成的。此H型机器人安装了具有3种系统的螺杆(1)／大间距的螺母(2)，能允许快速移动。它由轴承(6)通过执行机构M控制。三个绞接的平行四边形位于（4）的两端，在（5）的中间将螺母与水平平台（3）连接。机架（7）支撑着整个结构（图2）。边螺旋杆允许沿着其轴转动和移动。中心螺母则不允许平行四边形构架的转动。移动平台与半气缸相似，其自由度为3。这装置的主要优点是那工作空间是直接与平行轴长度成比例，能得到一个较大工作空间。

柱状-机器人

滑动对偶偶P较好的性有能在在工业机械元件上得到应用的可能。一个平行四边形能够利用四转动对偶偶R得到一个移动自由度。因此，利用柱状对偶偶代替平行四边形（Star 机器人）进行机器人设计是一个经济可行的方法。人们想象出了由CPR三重次序组成的很多几何排列（圆柱形对偶偶C可能能被RP代替以得到一电动千斤顶）。轴C必须在每次排列中与R轴平行。P对偶偶的方向可以是任意的。柱状机器人的草图见图3。两固定电动千斤顶是同轴的。第三个电动千斤顶为垂直安装。实际上，这些轴都是水平的。两柱状对偶偶相对偶于前两轴呈45度角。第三柱状对偶偶与第三轴垂直。移动平台在不需要人为调节的条件下在较大工作空间内自行移动。

结论

很多资料[10], [11], [12], [13], [14], [15]表明了假设群论的，特别是其动力学的重要性。通过对偶新的并行机器人的查证能够对偶我们进行机器人原型的构造有很大帮助。其机械性能的日益增加和制造费用的降低用使得机器人在当今工业制造中越来越具有吸引力。这种新机器人具有通用并行机器人在定位、灵敏性和马达定位安装方面的优点，可代替DELTA机器人。

简写列表 1

置换组的子群

{E} 恒等。

{t(D)} 对直线 D 的平移。

{R(N,u)} 绕轴旋转装置.( 或同等物对 N',和 NN 的 u'^u=O)

{H(N,u,p)} 转轴 (N ,u,p)= 2 k 的螺旋运动。

{t(P)} 对平面 P 的平移。

{C(N,u)} 沿轴平移的组合旋转装置.(N,u)

{t} 空间的平移。

{G(P)} 对平面P的平行平面运动。

{Y(w,p)} 平面垂直平移到 w 所允许的平移旋转和沿任何轴平行到 w 的旋转动作。{S(N)} 在点N周围的额球状的旋转装置。

{X(w)} 允许空间和沿任一轴旋转到 w 的平移旋转装置运动。

{D}综合刚体运动。

Design of parallel manipulators via the displacement group

Jacques M.HERVE

ECELE CENTRALE PARIS

92295 CHATENAY MALABRY CEDEX

FRANCE

Abstract:Our aim is to give a complete presentation of the application of Life Group Theory to the structural design of manipulator robots. We focused our attention on parallel manipulator robots and in particular those capable of spatial translation. This is justified by many industrial applications which do not need the orientation of the end-effectors in the space. The advantage of this method is that we can derive systematically all kinematics chains which produce the desired displacement subgroup. Hence, an entire family of robots results from our investigation. The

T-STAR manipulator is now a working device. H-ROBOT, PRISM-ROBOT are new possible robots. These manipulators respond to the increasing demand of fast working rhythms in modern production at a low cost and are suited for any kind of pick and place jobs like sorting, arranging on palettes, packing and assembly.

Keywords: Kinematics, Parallel Robot.

Introduction

The mathematical theory of groups can be applied to the set of displacements. If we can call {D} the set of all possible displacements, it is proved, according to this theory, that {D} have a group structure. The most remarkable movements of a rigid body are then represented by subgroups of {D}. This method leads to a classification of mechanism [1]. The main step for establishing such a classification is the derivation

of an exhaustive inventory of the subgroups of the displacement group. This can be done by a direct reasoning by examining all the kinds of products of rotations and translations [2].

However, a much more effective method consists in using Lie Group Theory [3] , [4]. Lie Groups are defined by analytical transformations depending on a finite number of real parameters. The displacement group {D} is a special case of a Lie Group of dimension six.

Lie’s Theory

Within the framework of Lie’ Theory, we associate infinitesimal transformations making

up a Lie algebra with finite operations which are obtained from the previous ones by exponentiation. Continuous analytical groups are described by the exponential of

differential operators which correspond to the infinitesimal transformations of the group.

Furthermore, group properties are interpreted by the algebraic structure of Lie algebra of the differential operators and conversely. We recall the main definition axiom of a Lie algebra: a Lie algebra is a vector space endowed with a bilinear skew symmetric closed product. It is well know [5] , that the set of screw velocity fields is a vector space of dimension six for the natural operations at a given point N.

By following the steps indicated in [3] we can produce the exhaustive list of the Lie subgroup of Euclidean displacements {D} (see synoptical list 1). This is done by first defining a differential operator associated with the velocity field. Then, by exponentiation, we derive the formal Lie expression of finite displacements which are shown to be equivalent to affine direct orthonormal transformations. Lie sub-algebras of screw velocity fields lead to the description of the displacement subgroups.

The {X (w)} subgroup

In order to generate spatial translation with parallel mechanisms, we are led to look for displacements subgroups the intersection of which is the spatial translation subgroup {T}.We will consider only the cases for which the intersection subgroup is strictly included in the two “parallel” subgroups. The most important case of this sort is the parallel association of two {X (w)} subgroups with two distinct vector directions w and w’. It is easy to prove:

{X(w)} {X(w’)}={T},w≠w’

The subgroup {X (w)} plays a prominent role in mechanism design. This subgroup combines spatial translation with rotation about a movable axis which remains parallel to given direction w , well defined by the unit vector w. Physical implementations of {X(w)} mechanical liaisons can be obtained by ordering in series kinematics pairs represented by subgroups of {X(w)}. Practically only prismatic pair and a revolute pair P, R, H are use to build robots (the cylindric pair C combines in a compact way a prismatic pair and a revolute pair). A complete list of all possible combinations of these kinematics pairs generating the {X (w)} subgroup is given in [6].

Two geometrical conditions have to be satisfied in the series: the rotation axes and the screw axes are parallel to the given vector w; there is no passive mobility.

The displacement operator for the {X {w}} subgroup, acting on point M is: M →N + ａu + bv + cw +exp(hw^) N M

^ is the symbol of the vector product.

Point N and the vectors u, v, w make up an orthogonal frame of reference in the space and a, b, c, h are the four parameters of the subgroup which has the dimension 4. Parallel robots for spatial translation

To produce spatial translation it is sufficient to place two mechanical generators of the subgroups {X(w)} and {X(w’)},w≠w’, in parallel, between a mobile platform and a fixed motors then three generators of the three subgroups

{X(w)},{X(w’)},{X(w’’)},w≠w’, is needed. Any series of P, R or H pairs which constitute a mechanical generator of the {X (w)} subgroup can be implemented. Morever, these three mechanical generators may be different or the same depending on the desired kinematics results. This wide range of combinations gives rise to an entire family of robots capable of spatial translation. Simulation of the most interesting architectures can easily be achieved and the choice of the robot to be constructed can therefore meet the needs of the commissioner.

Clavel’s Delta robot belongs to this family as it is based on the same kinematics principles [7].

The parallel manipulator Y-STAR

STAR [16] is made up by three cooperating arms which generate the subgroups {X (u)}, {X (u’)}, {X(u’’)}, (fig 1). The three arms are identical and each one generates a subgroup {X(u)} by the series RHPaR where Pa represents the circular translation liaison determined by the two opposite bars of a planar hinged parallelogram. The axes of the two revolute pairs and of the screw pair must be parallel in order to generate a {X (u)}, subgroup. For each arm, the first two pairs, i.e. the coaxial revolute pair and the screw pair, constitute the fixed part of the robot and form at the same time the mechanical structure of an axes lie on the same plane and divide it into three identical parts thus forming a Y shape. Hence the angle between any two axes is always 2 /3.The mobile part of the robot is made up by three PaR series that all converge to a common point below which the mobile platform is located. The platform stays parallel to the reference plane and cannot rotate about the axis perpendicular to this plane. Any kind of appropriate end effectors can be placed on this mobile platform.

The derivation of the {T} subgroup, which proves the mobile platform can only translate in the space, is given in [8].

The H – Robot

For a great majority of parallel robots including the Delta Robot and the Y Star, the working volume of the end effectors is small relative to the bulkiness of the whole device. It is the essential drawback of such a kind of manipulator. In order to avoid this native narrowness of the working volume, it can be imagine to implement three input electric jacks with three parallel axes instead of converging axes. Three arms similar to those of the Y Star cannot be employed: the intersection set of three equal set {X (v)} will be equal to {X (v)} instead of {T}. Hence, designing the new

H-Robot [16], we have chosen two arms of the Y Star type and a third pattern which may be compared with the Delta arms.

This third mechanism begins from the fixed frame with a motorized prismatic pair parallel to the first two electric jacks. It is followed by a hinged planar parallelogram which is free to rotate around an axis perpendicular to the P pair thanks to a bar of the parallelogram. The opposite bar is connected to the mobile platform via a revolute pair R of parallel axis. This property is maintained when the parallelogram changes of shape (with one degree of freedom).

In a first prototype built at “IUT de Ville D’Avray ”(France) by a team a students directed by the professor Pastoré, a H-Robot implements 3 systems screws (1) / nut (2) with a large pitch , which allow rapid movements. It is hold by bearings (6) and animated by the actuators M. Three planar hinged parallelograms, on both sides (4) and at the center (5) make the connection from the nuts to the horizontal platform (3). The stand (7) supports the whole structure (fig 2).

The side screws permit rotation and translation along their axes. The central nut does not allow the rotation of the parallelogram plane about the screw axis.

The mobile platform can only translate with 3 degrees of freedom inside the working space which may be assimilated to a half-cylinder.

The main advantage of this device is that the working volume is directly proportional to the length of the parallel axes and it can be made considerably large.

The Prism- Robot

Sliding pairs P of good quality are available in the industry of mechanical components.

A parallelogram employs four revolute pair R to generate a one degree of freedom translation motion. Therefore, the idea of implementing prismatic pairs instead of parallelograms (Star-Robot) seems to be an economic hint for a new robot design. Various geometric arrangements of three sequences CPR can be imagined (the cylindric pair C may be replaced by RP in order to make up an electric jack). The axis of C have to be parallel to the R axis in each sequence. The direction of P pair may be anyone. A selected sketch is the Prism-Robot of figure 3. Two fixed electric jacks are coaxial. A third fixes electric jack is perpendicular. For practical manipulators, these axes will be horizontal. Two prismatic pair are inclined with the angle 45o relative to the first two axes. The third prismatic pair will be perpendicular to the third axis. The mobile platform is able to undergo pure translation in a wide volume with no jamming effect.

Conclusions

The importance of Lie group theory, expecially for kinematics is recognized from various source [10], [11], [12], [13], [14], [15]. Investigation of new parallel robots generating pure translation led us to the construction of several prototypes. Increasing

performances and the low cost of fabrication make these robots attractive for modern industry. They are presented as an alternative to the DELTA robot and have the classical parallel robot advantages for positioning, precision, rapidity and fixed motor location.

References

[1] HERVE J.M, “Analyse structurelle des mécanismes par group des déplacements”, Mech, Mach, Theory 13, pp, 437-450 (1978).

[2] FRANGHELLA P, “Kinematics of Spatial Linkage by Group Algebra: a strucrure based approach”, Mech, Mach, Theory 23, no 3 pp, 171-183 (1988).

[3] HERVE J.M, “The mathematical group structure of the set of displacements”, Mech, Mach, Theory 29, no 1 pp, 71-83 (1994).

[4] HERVE J.M, “Intrinsic formulation of problems of geometry and kinematics of mechanism”, Mech, Mach, Theory 17, pp 179-184 (1994).

[5] SUGIMOTO K, DUFFY J, “Application of linear Algebra to Screw Systems”, Mech, Mach, Theory 17, pp, 73-83 (1994).

[6] HERVE J.M, SPARACINO F, “Structural Synthesis of Parallel Robots Generating Spatial Translation” 5th Int. Conf, on Adv, Robotics, IEEE no 91TH0367-4, V ol 1, pp. 808-813, 1991.

[7] CLA VEL R, “Delta, a fast robot with parallel geometry”, Proc. Int, Symp, on Industrial Robots, April 1988, pp 91-100.

[8] HERVE J.M, SPARACINO F, “Star, a New Concept in Robotics”, 3rd Intern. Workshop on Advances in Robot Kinematics, Sept. 7-9, 1992, Ferrara, Italy pp. 176-183.

[9] MERLET J.P, “Les robots parallèles”, Hermès, Paris,1990.

[10] KARGER A, NOV ACK J. Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, 1985.

[11] CHEVALLIER D.P, “Lie Algebras, Modules, Dual Quaternions and Algebraic Methods in Kinematics”, Mech, Mach, Theory, V ol. 26, no 6, pp, 613-627 (1994).

[12] POPPLESTONE R.J, “Group Theory and Robotics”, in Robotics Research. The First Int. Symp, M.Brady and R.Paul Eds, Cambrige, MM.MIT Press 1984.

[13] ANGELES J, “Spatial kinematics chains”, Springer Verlag, Berlin, 1982.

[14] HILLER M,WOERNIE C, “A Unified Repre sentation of Spatial Displacements”, Mech, Mach, Theory Vol. 19, pp, 477-486 (1984).

[15] SAMUEL A.E, Mc AREE P.R, HUNT K. H, “Unifying Screw Geometry and Matrix Transformations”, The International Journal of Robotics Research, V ol. 10, no5, October 1991.

[16] HERVE J.M, “Dispositif pour le déplacement en translation spatiale d’un element dans I’espace, en particulier pour robot mécanique”, French patent no 9100286 of January 11, 1991. European patent no 91403521.7 of December 23, 1991.

Synoptical list 1

Subgroups of the displacement group

{E} identity.

{T(D)} translations parallel to the straight line D.

{R(N,u)} rotations around the axis determined by the pair N,u (or and equivalent pair N’,u with NN’^u=O).

{H(N,u,p)} screw motions with the axis N,u and the pitch p = 2 k.

{T(P)} translations parallel to the plane P.

{C(N,u)} combined rotations and translations along an axis (N,u).

{T} spatial translation.

{G(P)} planar movements parallel to the plane P.

{Y(w,p)} screw translations allowing plane translations perpendicular to w and screw motions of pitch p along any axis parallel to w.

{S(N)} spheric rotations around the point N.

{X(w)} translating hinge motions allowing spatial translations and rotations around any axis parallel to w.

{D} general rigid body motions.