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附錄：
并聯(lián)位移機器人的設計
Jacques M.HERVE
ECELE CENTRALE PARIS
92295 CHATENAY MALABRY CEDEX
FRANCE
摘要:本文目的是對偶具有人性化機器人的應用做一個完全的介紹,并將著重討論并行機器人特別是那些能夠進行空間平移的機器人。在許多工業(yè)的應用過程中這種機器人被證明其末端執(zhí)行器在空間上的定位是沒必要的。這個方法的優(yōu)點是我們能系統(tǒng)地導出能預期得到位移子群的所有運動學鏈。因此，我們調查了機器人的整個家族。T-STAR機器人現(xiàn)在就是一臺工作裝置。而H-ROBOT,PRISM-ROBOT是新的可能的機器人。這些機器人能滿足現(xiàn)代生產快節(jié)奏工作中價格低以及符合挑選的工作環(huán)境，如選料、安排、包裝、裝配等發(fā)日益增長的需求。
關鍵詞：運動學,并行機器人
引言
群論可以運用于一系列位移當中。根據這個理論，如果我們能夠證明群{D}包含所有的可能的位移，那么{D}就具有群結構。剛體的最顯著運動是由群{D}表現(xiàn)出來的。這方法導致機械裝置的分類 [1]。建立這樣的一個分類的主要的步驟是將位移群的所有子群導出。這能通過檢驗所有具有旋轉和平移特性的[2]產品直接推理出。然而,一個更有效的方法存在于假設群論[3],[4]中。假設群論是在取決于許多有限實參數的全純映射的基礎上定義的。位移群{D}是六維假設群的一個特例。
假設理論
在假設群論的框架內,我們將用于補償李代數的微元變換與通過其前面冪運算得到的有限運算結合起來。連續(xù)群通過與群微元變換有關的微分冪運算描述出來。
另外，群體特性通過微分運算及其逆運算所得到的李代數的代數結構而得到了解釋。讓我們回憶一下李代數主要的定義公理：一個李代數是一個具有封閉乘積的反對偶稱雙線性的矢量空間。眾所周知 [5]，螺旋速度場是在給定點N的條件下通過運算得到的一個六維的矢量空間。由下面［3］中步驟表明，我們能得完整的歐幾里得位移{D}子群列表（見大綱表1）。該列表是通過首先定義一個與速度場有關的微分運算符得到的。然后，通過冪運算，得到了李代數有限位移的表達式。此表達式相當于仿射的直接歸一正交變換。螺旋速度場的子李代數是對偶位移子群組的直接描述。
{X (w)}子群
為了利用平行機理得到空間平移，我們需要找到所有位移子群的交集——空間平移子群{T}。我們考慮的子群交集將嚴格的包含于兩個“平行”子群內。此類別的最重要的情況是2個{X (w)} 子群和2個不同矢量方向w和w’的平行關系。這很容易證明：
{X(w)} {X(w’)}={T},w≠w’
子群{X (w)}在機制設計起一個很重要的作用。該子群由帶有旋轉運動的空間平移組成,其旋轉主軸方向與所給定的矢量w的方向始終平行。{X(w)}機械聯(lián)系的實際實施是通過子群{X(w)}代表的系列運動學對偶中的命令實現(xiàn)的。實際上棱柱對偶和旋轉對偶P,R,H都用于構造機器人(圓柱體對偶C以緊湊的方式結合棱柱對偶和旋轉對偶)。產生的這些運動學對偶的所有可能組合由子群組{X (w)}在[6]中給出。
同時它們必須連續(xù)的滿足兩種幾何情況：旋轉軸與螺旋軸要與給定的矢量w平行；不是被動運動。
{X{w}}子群的位移運算符,在M點的作用是:
M → N + ａu + bv + cw +exp(hw^) N M
^是矢量乘積標志。
點N和矢量u,v,w組成了空間的正交標架的基準。a, b, c, h為具有四維空間的子群的四個參數。
空間平移的并聯(lián)機器人
當兩子群組{X(w)} 和{X(w’)},w≠w’,滿足w≠w’，但矢量平行時，在移動平臺和固定馬達之間，其機械生成元就足以能產生空間平移。三個子群組{X (w)},{X(w’)},{X(w’’)},w≠w’時其生成元同樣也能產生空間平移。P，R或H的任何系列組成群組{X (w)}生成元的對偶的空間平移都能被實現(xiàn)。此外，這3種機械生成元可以是不同或一樣但都取決于所需的運動學結果。這種組合范圍很廣，使得整個能進行空間平移的機器人家族成員得到了增加。最有趣的是建筑的模擬能容易地是完成，機器手的選擇也能適應委員的需要。Clavel的Delta機器人屬于這個家族，因為它基于相同的運動學原理[7]。
并行操作機器人Y-STAR
STAR [16] 由3個能產生{X (u)}, {X (u’)}, {X(u’’)} (fig 1)子群組的協(xié)作操作臂組成。3只機械臂是相同且每只都能通過一系列的RHPaR生成一個子群{X (u)}，其中Pa代表循環(huán)平移協(xié)作，此平移協(xié)作由一塊絞接的平行四邊形的兩對偶立的桿控制決定。
兩旋轉對偶軸與螺旋對偶軸必須平行以保證能生成{X (u)}子群組。每條機械臂，第一個2對偶,即同軸旋轉對偶和螺旋對偶組成固定機器人的固定部分，同時形成處于相同平面的軸的機械結構，將其分為三個相同部分，從而形成了Y行狀。因此任意兩軸之間的角度都占整個空間角度的2 /3。機器人的移動部分由PaR系列組成，都能集中于移動平臺做指定的某點位置。平臺與參考平面保持平行，不能繞垂直于參考平面的軸旋轉。任何的一種專有的末端執(zhí)行器都能是放置在這流動的平臺上。 所得到的反應移動平臺的{T}子群僅能在空間進行平移,在[8]中給出。
H型機器人
大部分并型機器人包括Delta機器人和Y Star機器人，其末端執(zhí)行器的工作空間與整個裝置相比較小。這是此類機器人的一個缺陷。為了避免這種工作空間的限制，對偶此裝置安裝具有平行軸的電動千斤頂。與Y Star相似的機器人臂不能使用：三個相同集{X (v)}的交集等于{X (v)}而不是{T}。因此，在計新的H機器人[16]時，我們選擇與Y-Sta相同的兩條手臂，第三條手臂可與Delta手臂相比。這第三條機械臂開始形成帶有與第一個兩電動千斤頂平行的機動化柱狀對偶的固定框架。繼以之絞接的二維平行四邊形，此四邊形由于其中一根桿的緣故能繞垂直于P對偶的軸轉動。與此桿相對偶的桿經由平行軸的旋轉對偶R被連結到移動平臺上。當平行四邊形形狀變化時，這個性質被保持(自由度為一)。此機器人的第一個樣機有一個團隊的學生在Pastoré教授的指導下于法國“IUT de Ville D’Avray”完成的。此H型機器人安裝了具有3種系統(tǒng)的螺桿(1)／大間距的螺母(2)，能允許快速移動。它由軸承(6)通過執(zhí)行機構M控制。三個絞接的平行四邊形位于（4）的兩端，在（5）的中間將螺母與水平平臺（3）連接。機架（7）支撐著整個結構（圖2）。邊螺旋桿允許沿著其軸轉動和移動。中心螺母則不允許平行四邊形構架的轉動。移動平臺與半氣缸相似，其自由度為3。這裝置的主要優(yōu)點是那工作空間是直接與平行軸長度成比例，能得到一個較大工作空間。
柱狀-機器人
滑動對偶偶P較好的性有能在在工業(yè)機械元件上得到應用的可能。一個平行四邊形能夠利用四轉動對偶偶R得到一個移動自由度。因此，利用柱狀對偶偶代替平行四邊形（Star機器人）進行機器人設計是一個經濟可行的方法。人們想象出了由CPR三重次序組成的很多幾何排列（圓柱形對偶偶C可能能被RP代替以得到一電動千斤頂）。軸C必須在每次排列中與R軸平行。P對偶偶的方向可以是任意的。柱狀機器人的草圖見圖3。兩固定電動千斤頂是同軸的。第三個電動千斤頂為垂直安裝。實際上，這些軸都是水平的。兩柱狀對偶偶相對偶于前兩軸呈45度角。第三柱狀對偶偶與第三軸垂直。移動平臺在不需要人為調節(jié)的條件下在較大工作空間內自行移動。
結論
很多資料[10], [11], [12], [13], [14], [15]表明了假設群論的，特別是其動力學的重要性。通過對偶新的并行機器人的查證能夠對偶我們進行機器人原型的構造有很大幫助。其機械性能的日益增加和制造費用的降低用使得機器人在當今工業(yè)制造中越來越具有吸引力。這種新機器人具有通用并行機器人在定位、靈敏性和馬達定位安裝方面的優(yōu)點，可代替DELTA機器人。
簡寫列表 1
置換組的子群
{E} 恒等。
{t(D)} 對直線 D 的平移。
{R(N,u)} 繞軸旋轉裝置.( 或同等物對 N',和 NN 的 u'^u=O)
{H(N,u,p)} 轉軸 (N ,u,p)= 2 k 的螺旋運動。
{t(P)} 對平面 P 的平移。
{C(N,u)} 沿軸平移的組合旋轉裝置.(N,u)
{t} 空間的平移。
{G(P)} 對平面P的平行平面運動。
{Y(w,p)} 平面垂直平移到 w 所允許的平移旋轉和沿任何軸平行到 w 的旋轉動作。
{S(N)} 在點N周圍的額球狀的旋轉裝置。
{X(w)} 允許空間和沿任一軸旋轉到 w 的平移旋轉裝置運動。
{D} 綜合剛體運動。
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, Vol 1, pp. 808-813, 1991.
[7] CLAVEL 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, NOVACK 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, Vol. 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, Vol. 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 = 2k.
{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.
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