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并聯(lián)位移機(jī)器人的設(shè)計(jì)
Jacques M.HERVE
ECELE CENTRALE PARIS
92295 CHATENAY MALABRY CEDEX
FRANCE
摘要:本文目的是對(duì)偶具有人性化機(jī)器人的應(yīng)用做一個(gè)完全的介紹,并將著重討論并行機(jī)器人特別是那些能夠進(jìn)行空間平移的機(jī)器人。在許多工業(yè)的應(yīng)用過程中這種機(jī)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(gè)方法的優(yōu)點(diǎn)是我們能系統(tǒng)地導(dǎo)出能預(yù)期得到位移子群的所有運(yùn)動(dòng)學(xué)鏈。因此，我們調(diào)查了機(jī)器人的整個(gè)家族。T-STAR機(jī)器人現(xiàn)在就是一臺(tái)工作裝置。而H-ROBOT,PRISM-ROBOT是新的可能的機(jī)器人。這些機(jī)器人能滿足現(xiàn)代生產(chǎn)快節(jié)奏工作中價(jià)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(zhǎng)的需求。
關(guān)鍵詞：運(yùn)動(dòng)學(xué),并行機(jī)器人
引言
群論可以運(yùn)用于一系列位移當(dāng)中。根據(jù)這個(gè)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(jié)構(gòu)。剛體的最顯著運(yùn)動(dòng)是由群{D}表現(xiàn)出來的。這方法導(dǎo)致機(jī)械裝置的分類 [1]。建立這樣的一個(gè)分類的主要的步驟是將位移群的所有子群導(dǎo)出。這能通過檢驗(yàn)所有具有旋轉(zhuǎn)和平移特性的[2]產(chǎn)品直接推理出。然而,一個(gè)更有效的方法存在于假設(shè)群論[3],[4]中。假設(shè)群論是在取決于許多有限實(shí)參數(shù)的全純映射的基礎(chǔ)上定義的。位移群{D}是六維假設(shè)群的一個(gè)特例。
假設(shè)理論
在假設(shè)群論的框架內(nèi),我們將用于補(bǔ)償李代數(shù)的微元變換與通過其前面冪運(yùn)算得到的有限運(yùn)算結(jié)合起來。連續(xù)群通過與群微元變換有關(guān)的微分冪運(yùn)算描述出來。
另外，群體特性通過微分運(yùn)算及其逆運(yùn)算所得到的李代數(shù)的代數(shù)結(jié)構(gòu)而得到了解釋。讓我們回憶一下李代數(shù)主要的定義公理：一個(gè)李代數(shù)是一個(gè)具有封閉乘積的反對(duì)偶稱雙線性的矢量空間。眾所周知 [5]，螺旋速度場(chǎng)是在給定點(diǎn)N的條件下通過運(yùn)算得到的一個(gè)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(gè)與速度場(chǎng)有關(guān)的微分運(yùn)算符得到的。然后，通過冪運(yùn)算，得到了李代數(shù)有限位移的表達(dá)式。此表達(dá)式相當(dāng)于仿射的直接歸一正交變換。螺旋速度場(chǎng)的子李代數(shù)是對(duì)偶位移子群組的直接描述。
{X (w)}子群
為了利用平行機(jī)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(yán)格的包含于兩個(gè)“平行”子群內(nèi)。此類別的最重要的情況是2個(gè){X (w)} 子群和2個(gè)不同矢量方向w和w’的平行關(guān)系。這很容易證明：
{X(w)} {X(w’)}={T},w≠w’
子群{X (w)}在機(jī)制設(shè)計(jì)起一個(gè)很重要的作用。該子群由帶有旋轉(zhuǎn)運(yùn)動(dòng)的空間平移組成,其旋轉(zhuǎn)主軸方向與所給定的矢量w的方向始終平行。{X(w)}機(jī)械聯(lián)系的實(shí)際實(shí)施是通過子群{X(w)}代表的系列運(yùn)動(dòng)學(xué)對(duì)偶中的命令實(shí)現(xiàn)的。實(shí)際上棱柱對(duì)偶和旋轉(zhuǎn)對(duì)偶P,R,H都用于構(gòu)造機(jī)器人(圓柱體對(duì)偶C以緊湊的方式結(jié)合棱柱對(duì)偶和旋轉(zhuǎn)對(duì)偶)。產(chǎn)生的這些運(yùn)動(dòng)學(xué)對(duì)偶的所有可能組合由子群組{X (w)}在[6]中給出。
同時(shí)它們必須連續(xù)的滿足兩種幾何情況：旋轉(zhuǎn)軸與螺旋軸要與給定的矢量w平行；不是被動(dòng)運(yùn)動(dòng)。
{X{w}}子群的位移運(yùn)算符,在M點(diǎn)的作用是:
M → N + ａu + bv + cw +exp(hw^) N M
^是矢量乘積標(biāo)志。
點(diǎn)N和矢量u,v,w組成了空間的正交標(biāo)架的基準(zhǔn)。a, b, c, h為具有四維空間的子群的四個(gè)參數(shù)。
空間平移的并聯(lián)機(jī)器人
當(dāng)兩子群組{X(w)} 和{X(w’)},w≠w’,滿足w≠w’，但矢量平行時(shí)，在移動(dòng)平臺(tái)和固定馬達(dá)之間，其機(jī)械生成元就足以能產(chǎn)生空間平移。三個(gè)子群組{X (w)},{X(w’)},{X(w’’)},w≠w’時(shí)其生成元同樣也能產(chǎn)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(duì)偶的空間平移都能被實(shí)現(xiàn)。此外，這3種機(jī)械生成元可以是不同或一樣但都取決于所需的運(yùn)動(dòng)學(xué)結(jié)果。這種組合范圍很廣，使得整個(gè)能進(jìn)行空間平移的機(jī)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(jī)器手的選擇也能適應(yīng)委員的需要。Clavel的Delta機(jī)器人屬于這個(gè)家族，因?yàn)樗谙嗤倪\(yùn)動(dòng)學(xué)原理[7]。
并行操作機(jī)器人Y-STAR
STAR [16] 由3個(gè)能產(chǎn)生{X (u)}, {X (u’)}, {X(u’’)} (fig 1)子群組的協(xié)作操作臂組成。3只機(jī)械臂是相同且每只都能通過一系列的RHPaR生成一個(gè)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(duì)偶立的桿控制決定。
兩旋轉(zhuǎn)對(duì)偶軸與螺旋對(duì)偶軸必須平行以保證能生成{X (u)}子群組。每條機(jī)械臂，第一個(gè)2對(duì)偶,即同軸旋轉(zhuǎn)對(duì)偶和螺旋對(duì)偶組成固定機(jī)器人的固定部分，同時(shí)形成處于相同平面的軸的機(jī)械結(jié)構(gòu)，將其分為三個(gè)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(gè)空間角度的2 /3。機(jī)器人的移動(dòng)部分由PaR系列組成，都能集中于移動(dòng)平臺(tái)做指定的某點(diǎn)位置。平臺(tái)與參考平面保持平行，不能繞垂直于參考平面的軸旋轉(zhuǎn)。任何的一種專有的末端執(zhí)行器都能是放置在這流動(dòng)的平臺(tái)上。 所得到的反應(yīng)移動(dòng)平臺(tái)的{T}子群僅能在空間進(jìn)行平移,在[8]中給出。
H型機(jī)器人
大部分并型機(jī)器人包括Delta機(jī)器人和Y Star機(jī)器人，其末端執(zhí)行器的工作空間與整個(gè)裝置相比較小。這是此類機(jī)器人的一個(gè)缺陷。為了避免這種工作空間的限制，對(duì)偶此裝置安裝具有平行軸的電動(dòng)千斤頂。與Y Star相似的機(jī)器人臂不能使用：三個(gè)相同集{X (v)}的交集等于{X (v)}而不是{T}。因此，在計(jì)新的H機(jī)器人[16]時(shí)，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(jī)械臂開始形成帶有與第一個(gè)兩電動(dòng)千斤頂平行的機(jī)動(dòng)化柱狀對(duì)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(duì)偶的軸轉(zhuǎn)動(dòng)。與此桿相對(duì)偶的桿經(jīng)由平行軸的旋轉(zhuǎn)對(duì)偶R被連結(jié)到移動(dòng)平臺(tái)上。當(dāng)平行四邊形形狀變化時(shí)，這個(gè)性質(zhì)被保持(自由度為一)。此機(jī)器人的第一個(gè)樣機(jī)有一個(gè)團(tuán)隊(duì)的學(xué)生在Pastoré教授的指導(dǎo)下于法國(guó)“IUT de Ville D’Avray”完成的。此H型機(jī)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(dòng)。它由軸承(6)通過執(zhí)行機(jī)構(gòu)M控制。三個(gè)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(tái)（3）連接。機(jī)架（7）支撐著整個(gè)結(jié)構(gòu)（圖2）。邊螺旋桿允許沿著其軸轉(zhuǎn)動(dòng)和移動(dòng)。中心螺母則不允許平行四邊形構(gòu)架的轉(zhuǎn)動(dòng)。移動(dòng)平臺(tái)與半氣缸相似，其自由度為3。這裝置的主要優(yōu)點(diǎn)是那工作空間是直接與平行軸長(zhǎng)度成比例，能得到一個(gè)較大工作空間。
柱狀-機(jī)器人
滑動(dòng)對(duì)偶偶P較好的性有能在在工業(yè)機(jī)械元件上得到應(yīng)用的可能。一個(gè)平行四邊形能夠利用四轉(zhuǎn)動(dòng)對(duì)偶偶R得到一個(gè)移動(dòng)自由度。因此，利用柱狀對(duì)偶偶代替平行四邊形（Star機(jī)器人）進(jìn)行機(jī)器人設(shè)計(jì)是一個(gè)經(jīng)濟(jì)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(duì)偶偶C可能能被RP代替以得到一電動(dòng)千斤頂）。軸C必須在每次排列中與R軸平行。P對(duì)偶偶的方向可以是任意的。柱狀機(jī)器人的草圖見圖3。兩固定電動(dòng)千斤頂是同軸的。第三個(gè)電動(dòng)千斤頂為垂直安裝。實(shí)際上，這些軸都是水平的。兩柱狀對(duì)偶偶相對(duì)偶于前兩軸呈45度角。第三柱狀對(duì)偶偶與第三軸垂直。移動(dòng)平臺(tái)在不需要人為調(diào)節(jié)的條件下在較大工作空間內(nèi)自行移動(dòng)。
結(jié)論
很多資料[10], [11], [12], [13], [14], [15]表明了假設(shè)群論的，特別是其動(dòng)力學(xué)的重要性。通過對(duì)偶新的并行機(jī)器人的查證能夠?qū)ε嘉覀冞M(jìn)行機(jī)器人原型的構(gòu)造有很大幫助。其機(jī)械性能的日益增加和制造費(fèi)用的降低用使得機(jī)器人在當(dāng)今工業(yè)制造中越來越具有吸引力。這種新機(jī)器人具有通用并行機(jī)器人在定位、靈敏性和馬達(dá)定位安裝方面的優(yōu)點(diǎn)，可代替DELTA機(jī)器人。
簡(jiǎn)寫列表 1
置換組的子群
{E} 恒等。
{t(D)} 對(duì)直線 D 的平移。
{R(N,u)} 繞軸旋轉(zhuǎn)裝置.( 或同等物對(duì) N',和 NN 的 u'^u=O)
{H(N,u,p)} 轉(zhuǎn)軸 (N ,u,p)= 2 k 的螺旋運(yùn)動(dòng)。
{t(P)} 對(duì)平面 P 的平移。
{C(N,u)} 沿軸平移的組合旋轉(zhuǎn)裝置.(N,u)
{t} 空間的平移。
{G(P)} 對(duì)平面P的平行平面運(yùn)動(dòng)。
{Y(w,p)} 平面垂直平移到 w 所允許的平移旋轉(zhuǎn)和沿任何軸平行到 w 的旋轉(zhuǎn)動(dòng)作。
{S(N)} 在點(diǎn)N周圍的額球狀的旋轉(zhuǎn)裝置。
{X(w)} 允許空間和沿任一軸旋轉(zhuǎn)到 w 的平移旋轉(zhuǎn)裝置運(yùn)動(dòng)。
{D} 綜合剛體運(yùn)動(dòng)。
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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