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modes differ from the ones of the 3-RPS **parallel** **manipulator** discussed in [2]. 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 [16], [18]. One such **manipulator** was already successfully built and proved the viability of using pantographs [19].

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Keywords: **parallel** **manipulator**, dual reconfigurable, 4-RUU, operation modes, **workspace**
1 Introduction
Different configuration space regions of **a** **parallel** **manipulator** (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. [1] 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 [5]. 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 [6]. 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.

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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-

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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

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1 INTRODUCTION
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 [4] occur in the **workspace** where the moving platform cannot resist any effort. They are very undesirable and generally eliminated by design. Serial singularities [5] 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 [6]. 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.

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Abstract
**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.

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I. I NTRODUCTION
There are many types of reconfigurable **parallel** manipu- lators. This paper deals with the determination of **a** control scheme **for** **a** reconfigurable **parallel** **manipulator** 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 **workspace** **for** **a** 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 [1]. The kinetostatic performance of **a** planar **parallel** mechanism, NaVARo, with variable actuation modes is presented in [2], [3]. The homing, calibration and model-based predictive control **for** the 4RPR **parallel** architecture is shown in [4].

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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 **workspace** **for** 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 **a** **manipulator** whose **workspace** contains **a** prescribed **workspace** and the sec- ond one aims to design **a** **manipulator**, of which the **workspace** is as large as possible. However, maximizing the **manipulator** **workspace** 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**.

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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 **a** **parallel** 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 [10] **for** **a** Gough-Stewart platform [14, 37]). Much of the work is thus devoted to solving this problem accurately and in real-time (see **for** instance [39]), or to **designing** **parallel** 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 [6], which simplifies the kinematic models by introducing additional sensors into the mechanism and thus yields easier control [26].

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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 [1], 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.

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Octree models are hierarchical data structures based on recursive subdivision of the space, respectively [16]. They are useful **for** representing complex 3 dimensional shapes like workspaces [10]. **A** close method is used in [17] that divides the **workspace** into boxes. This method does not use recursive subdivision but interval analysis methods [18] 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.

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Keywords: design, modeling, singularity, 3-DoF planar **parallel** **manipulator**, unlimited rotation capability. 1
I Introduction
In the search **for** **a** 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 [1], [2].

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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.

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the prototype.
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 **for** **a** two- degree of freedom translational **parallel** **manipulator**, 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.

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Keywords: **parallel** **manipulator**, epicyclic system,
kinematics, **workspace**
1 Introduction
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 [12] and the McGill Sch¨onflies-motion generator [13]. There are also five-dof **parallel** manipulators, such as 3T2R **parallel** manip- ulators [14, 15]. **For** six-dof manipulators, the number of all

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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 [7]. Several studies were performed by inventors to determine the geometric model, the usable **workspace** and the forces into the machine components [8]. The work presented in this paper is about stiffness modeling of H4 robot and can be easily extended to lower manipulability **parallel**

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