modes differ from the ones of the 3-RPS parallelmanipulator discussed in . The orientation workspace is parametrized to recognize the physical interpretations of rotational motions. The singularities are investigated and a performance index, named Maximum Inscribed Circle Di- ameter (MICD), is proposed. This index is used for the dimension synthesis of the manipulator studied in this paper. The reconfiguration strategy is presented to ensure that the manipulator can work at minimum required orientation for any moving-platform height.
This paper is the first to provide such a solution through the use of pantographs. Of course, the proposed design is more complicated than the simple Quadrupteron or Isoglide4 of Fig. 1, but this seems to be a reasonable price to pay. Moreover, the new robot is only made of three identical legs, in contrast to other 3T1R decoupled parallel robots, which is a great advantage in terms of workspace volume and acceleration capacities. Indeed, the proposed design is the result of a large study on the synthesis of parallel manipulators using pantographs , . One such manipulator was already successfully built and proved the viability of using pantographs .
Keywords: parallelmanipulator, dual reconfigurable, 4-RUU, operation modes, workspace
Different configuration space regions of aparallelmanipulator (PM) with possi- bly different type or number of degrees of freedom are known as the operation modes. They were first observed for the 3-URU DYMO robot by Zlatanov et al.  showing that it exhibits five different types of platform motion. A com- plete characterization of operation modes for various PMs using Study’s kine- matic mapping and algebraic geometry tools has been accomplished in [2–4]. Moreover, the operation modes of a much more complicated 4-RUU PM with vertical R-joint axes were determined by splitting it into two 2-RUU PMs . Additionally, a novel dual reconfigurable 4-rRUU PM was proposed, constituting a double-Hooke’s joint in each limb to reconfigure the R-joint axes to have any horizontal orientations. Its operation modes were determined for some specific configurations . This paper proposes an interpolation approach to determine the operation modes as a function of the orientation of base R-joint axes. Fur- thermore, its workspace is plotted, thanks to the algebraic expression of its limb singularities.
However, the existing PKM suffer from two major draw- backs, namely, a complex Cartesian workspace and highly non linear input/output relations. For most PKM, the Jacobian ma- trix which relates the joint rates to the output velocities is not constant and not isotropic. Consequently, the performances (e.g. maximum speeds, forces accuracy and rigidity) vary consider- ably for different points in the Cartesian workspace and for dif- ferent directions at one given point. This is a serious drawback for machining applications (Kim (1997); Treib et al. (1998); Wenger et al. (1999)). To be of interest for machining applica- tions, a PKM should preserve good workspace properties, that is, regular shape and acceptable kinetostatic performances through- out. In milling applications, the machining conditions must re- main constant along the whole tool path (Rehsteiner (1999); Rehsteiner et al. (1999)). In many research papers, this crite-
have nice interesting properties like the existence of an implicit adjacency graph which enables simple connectivity analyses (Wenger, 97) (Samet,79). An efficient calculation technique using octrees has been developed by the authors (Chablat, 97) but is not reported here for lack of space. The dimensions and the displace- ment ranges of the linear actuators are such that the dextrous workspace is non zero. That is, the output link can admit any orientation. In other words, the upper and lower sides of the workspace actually coincide (the workspace has the structure of a torus). Another important feature is that a singular surface lies in- side this workspace and divides it into two adjacent aspects (Sefrioui, 92). When the manipulator output link lies on this singular surface, the manipulator is in a configuration such that the axes on the linear actuators intersect at a common point. Figure 3 shows the singular surface and the two aspects are depicted in
The kinematic design of parallel mechanism has drawn the interest of several researchers. The workspace is usually considered as a relevant design criterion [1, 2, 3]. Parallel singularities  occur in the workspace where the moving platform cannot resist any effort. They are very undesirable and generally eliminated by design. Serial singularities  occur if the mechanism admit several solutions to the inverse kinematic model. To cope with the existence of multiple inverse kinematic solutions in serial manipulators, the notion of aspects was introduced in . The aspects equal the maximal singularity-free domains in the joint space. For usual industrial serial manipulators, the aspects were found to be the maximal sets in the joint space where there is only one inverse kinematic solution.
Parallel robots have proved they can have better per- formances than serial ones in term of rigidity and payload-to-weight ratio. Nevertheless their workspace is largely reduced by the presence of singularities. In particular, the Type 2 singularities (parallel singulari- ties) separate the workspace in different aspects, cor- responding to one (or more) robot assembly modes. In order to enlarge the workspace size, it has been proved that a mechanism can cross the singularity loci by using an optimal motion planning. However, if the trajectory is not robust to modeling errors, the robot can stop in the singularity and stay blocked.
I. I NTRODUCTION
There are many types of reconfigurable parallel manipu- lators. This paper deals with the determination of a control scheme fora reconfigurable parallelmanipulator with different actuation schemes. It is well known that the kinetostatic per- formance of parallel manipulators may change drastically in the neighborhood of parallel singularities. Parallel mechanisms may have several solutions to their inverse kinematics problem, i.e. several working modes. It is usually difficult to get a large singularity-free workspacefora given working mode. Hence, one must plan the trajectory in such a way that there should be change of working mode to avoid the singularities. These problems can be solved by introducing actuation redundancy, which involves force control algorithms . The kinetostatic performance of a planar parallel mechanism, NaVARo, with variable actuation modes is presented in , . The homing, calibration and model-based predictive control for the 4RPR parallel architecture is shown in .
Regular workspace size The quality of the manipula- tor workspace is of prime importance for the design of Parallel Kinematics Machines (PKMs). It is partly characterized by its size and shape. Moreover, the lower the amount of singularities throughout the workspace, the better the workspacefor continu- ous trajectory planning. The workspace optimization of parallel manipulators can usually be solved by means of two different formulations. The first formulation aims to design amanipulator whose workspace contains a prescribed workspace and the sec- ond one aims to design amanipulator, of which the workspace is as large as possible. However, maximizing the manipulatorworkspace may result in a poor design with regard to the manip- ulator dexterity and manipulability [12, 19]. This problem can be solved by properly defining the constraints of the optimiza- tion problem. Here, the multiobjective optimization problem of PPMs is based on the formulation of workspace maximization, i.e, the determination of the optimum geometric parameters in order to maximize a regular-shaped workspace.
Figure 1: A Gough-Stewart platform.
Consequently, one needs to be able to estimate or measure the latter. As far as we know, all the effort has been put on the estimation of the end-effector pose through the forward kinematic model and the joint measurements. How- ever, this yields much trouble, related to the fact that there is usually no an- alytic formulation of the forward kinematic model of aparallel mechanism. Hence, one numerically inverts the inverse kinematic model, which is analyti- cally defined for most of the parallel mechanisms. However, it is known [28, 16] that this numerical inversion requires high order polynomial root determination, with several possible solutions (up to 40 real solutions  fora Gough-Stewart platform [14, 37]). Much of the work is thus devoted to solving this problem accurately and in real-time (see for instance ), or to designingparallel mech- anisms with analytical forward kinematic model [18, 13]. Alternately, one of the promising paths lies in the use of the so-called metrological redundancy , which simplifies the kinematic models by introducing additional sensors into the mechanism and thus yields easier control .
algebraic approach called ‘Study kinematic mapping’ of the Eu- clidean group SE(3). The mechanism was described by using a set of eight constraint equations. A primary decomposition over a set of constraint equations revealed the existence of two op- eration modes. Singularity conditions were also determined by calculating the determinant of the Jacobian matrix with respect to the Study parameters. Following the research conducted in , this paper focuses on the study of the force/motion transmission capabilities of the two operation modes of the 3-PRS parallel ma- nipulator for different values of angle α. Output and constraint transmission indices are computed to evaluate the force/motion transmission capability in each operation mode. Eventually, the performance of the two operation modes can be compared and analysed within the orientation workspace. Some manipulator poses are presented to visualize the effectiveness of these indices. This paper is organised as follows: Section 2 presents a brief review on performance indices associated with output and con- straint transmission indices. Section 3 presents the architecture of the manipulator and its operation modes. In section 4, the Ja- cobian matrix of the 3-PRS manipulator is derived based on the Screw theory . In section 5, performance comparison of the two operation modes is discussed based on the distribution of output and constraint transmission indices in the orientation workspace. Eventually, section 6 discusses some of the singular configura- tions of the manipulator in the two operation modes.
Octree models are hierarchical data structures based on recursive subdivision of the space, respectively . They are useful for representing complex 3 dimensional shapes like workspaces . A close method is used in  that divides the workspace into boxes. This method does not use recursive subdivision but interval analysis methods  to build the dextrous workspace. However, it does not make it possible to perform Boolean operations or to make path-connectivity analysis easily. The first method permits us to calculate easily all kind of space and the computing time is limited as a function of the accuracy. The second one is more exact but requires more ability to be implemented. In both cases, we can characterize spaces whose dimensions are either lengths or angles.
Keywords: design, modeling, singularity, 3-DoF planar parallelmanipulator, unlimited rotation capability. 1
In the search fora suitable means for simulating flight conditions for the safe training of helicopter pilots, the design of parallel mechanisms has been proposed having all the freedom of motion and capable of being controlled in all of them simultaneously. A typical 6-DoF parallel mechanism consists of a moving platform connected to a fixed base by six serial chains called legs or limbs. Due to its parallel structure these manipulators offer the advantages of low inertia, considerable stiffness, large payload to manipulator weight ratio and higher operating speeds. The advantages however come at the expense of a reduced workspace, difficult mechanical design and more complex kinematics and control algorithms. They have been utilized for many practical applications and many researchers have paid attention to the design of these structures. Therefore, a large number of papers place emphasis on the study of 6-DoF parallel manipulators , .
A six DOF epicyclic parallelmanipulator, Monash Epicyclic-ParallelManipulator (MEPaM), is presented in  with all actuators mounted on the base, parallel singularity is independent on the position of the end-effector. Non-singular assembly mode changing of a six DOF parallelmanipulator 3- PPPS manipulator is shown in . Eight solutions to its DKP, several assembly modes can be connected by a non singular tra- jectories by encircle the cusp points in the joint space. The sin- gularity analysis of a six dof three-legged parallelmanipulatorfor force feedback interface, using Grassmann-Cayley algebra jacobian and Gr¨obner basis , shown in . The cylindrical algebraic decomposition (CAD) algorithm  is used to study the workspace and joint space, and a Gr¨obner based elimination process is used to compute the parallel singularities of the ma- nipulator [21–23].
In general, a robot design process has to simultaneously deals with the kinematic and kineto- static/dynamic aspects, both of which include a number of performance measures that essentially vary throughout the workspace. This can be effectively achieved by virtue of multiobjective optimization method. The multiobjective optimization problems of parallel manipulators (PMs) have been reported in the literature, where various approaches of multiobjective optimization have been applied to different types of PMs, while considering kinematic, dynamic and static criteria [11, 15, 16, 17, 18]. However, a systematic approach lacks in the optimum design for this class of SPMs, as the static/dynamic performance received relatively less attention as mentioned above.
The goal of constructing a prototype of the 4-rRUU PM is to use it for milling operations. A milling cutter will be mounted on the moving platform, whose axis will be normal to the latter as roughly represented in Fig. 5 (not drawn to scale). The workpiece is assumed to be a cuboid of dimension 0.12 m × 0.05 m × 0.05 m. To place the workpiece, it is necessary choose a location in the workspace that is free of internal collisions and far from singularities and is done as follows:
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This paper presents a design procedure fora two- degree of freedom translational parallelmanipulator, named IRSBot-2. This design procedure aims at determining the optimal design parameters of the IRSBot-2 such that the robot can reach a velocity equal to 6 m/s, an acceleration up to 20 G and a multi-directional repeatability up to 20 µm throughout its operational workspace. Besides, contrary to its counterparts, the stiffness of the IRSBot-2 should be very high along the normal to the plane of motion of its moving- platform. A semi-industrial prototype of the IRSBot-2 has been realized based on the obtained optimum design param- eters. This prototype and its main components are described in the paper. Its accuracy, repeatability, elasto-static per- formance, dynamic performance and elasto-dynamic perfor- mance have been measured and analyzed as well. It turns out the IRSBot-2 has globally reached the prescribed spec- ifications and is a good candidate to perform very fast and accurate pick-and-place operations.
Keywords: parallelmanipulator, epicyclic system,
There are a large number of parallel manipulators which have been reported over the past three decades. They can be divided into three-dof, four-dof, five-dof and six-dof parallel manipulators [1–4]. In three-dof parallel manipulators, there are three-dof translational parallel manipulators [5–7], three- dof rotational parallel manipulators [8, 9] and others [10, 11]. Among the four-dof parallel manipulators, examples include the four-DOF four-URU parallel mechanism  and the McGill Sch¨onflies-motion generator . There are also five-dof parallel manipulators, such as 3T2R parallel manip- ulators [14, 15]. For six-dof manipulators, the number of all
Stiffness is a crucially important performance specifica- tion of parallel kinematic machines. In order to get a real industrial machine, the H4 prototype (Fig. 1) must be opti- mized. Because of its long arms and rods, designers must be particularly careful with the machine stiffness, which has direct consequences on manipulation accuracy . Several studies were performed by inventors to determine the geometric model, the usable workspace and the forces into the machine components . The work presented in this paper is about stiffness modeling of H4 robot and can be easily extended to lower manipulability parallel