【正文】 三軸垂直。 結(jié)論 很多資料 [10], [11], [12], [13], [14], [15]表明了假設(shè)群論的，特別是其動力學(xué)的重要性。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。 20 簡寫列表 1 置換組的子群 {E} 恒等。 {R(N,u)} 繞軸旋轉(zhuǎn)裝置 .( 或同等物對 N39。^u=O) {H(N,u,p)} 轉(zhuǎn)軸 (N ,u,p)= 2 k 的螺旋運(yùn)動。 {C(N,u)} 沿軸平移的組合旋轉(zhuǎn)裝置 .(N,u) {t} 空間的平移。 {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)動作。 {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。 Design of parallel manipulators via the displacement group Jacques ECELE CENTRALE PARIS 92295 CHATENAY MALABRY CEDEX FRANCE Abstract: Our aim is to give a plete 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 endeffectors 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 TSTAR manipulator is now a working device. HROBOT, PRISMROBOT 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 21 {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 subalgebras 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 bines 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 22 build robots (the cylindric pair C bines in a pact way a prismatic pair and a revolute pair). A plete list of all possible binations 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。, a HRobot 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 halfcylinder. The main advantage of this device is that the working volume is directly