Vigen Arakelian
email: vigen.arakelyan@insa-rennes.fr

Sebastien Briot
email: sebastien.briot@ens.insa-rennes.fr

Improvement of Positioning Accuracy of PAMINSA (Parallel Manipulator of the I.N.S.A.)

In this paper we present new design approaches for the positional accuracy improvement of the 4-DOF parallel manipulator PAMINSA. The particularity of this manipulator is the decoupling of displacements in the horizontal plane from its translation along the vertical axis. The positioning errors caused by the inclination of the platform relating to the base are cancelled by means of two different approaches. The first solution is realized by means of two correcting systems mounted on the drive system of the vertical displacements. The second solution is carried out by a correcting mass mounted on the platform. The suggested solutions are illustrated by the numerical examples and the performances of such a design are shown.

Introduction

It is well known that the parallel manipulators show a better stiffness than the serial manipulators. Their actuators and gearing are fixed, resulting in a low moving mass. Thus, the payloads capacity of parallel manipulators with lightweight links is more important than the serial manipulators. However they have nonlinear coupled kinematics and dynamics, which is a drawback for the industrial applications. It is obvious that the parallel structure with linear inputoutput equations is more appealing than the nonlinear one. In recent years, the decoupling of motions of parallel manipulators has attracted researchers' attention and different structures have been proposed. Previous works on this problem may be arranged into two principal groups: (i) fully decoupled parallel structures in which the architecture of the manipulator is such that its input/output equations are linear and (ii) position/orientation decoupled manipulators in which the end-effector's position is independent of the orientation.

One of the first studies devoted to the design of fully decoupled parallel structures was developed by [START_REF] Kong | A class of 3-dof translational parallel manipulator with linear input/output equations[END_REF]. In this work, a class of 3-legged 3-DOF translational parallel manipulators with linear input-output equations was proposed. Another simple fully decoupled 3-DOF translational parallel mechanism named Tripteron was discussed in the paper [START_REF] Gosselin | A fully decoupled 3-dof translational parallel mechanism[END_REF]. Carricato and Parenti-Castelli (2004) proposed a novel parallel mechanism with fully decoupled degrees of freedom. This mechanism consists of two interconnected slider crank linkages, each of which independently actuates one of the Euler angles of the input link.

Several types of translational parallel mechanisms with decoupled motions were discussed in the next work of the same authors ( 2004). The parallel robot, called T3R1, with four degrees of freedom and decoupled motions was developed at the French Institute of Advanced Mechanics [START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot with uncoupled kinematics[END_REF].

In spite of rather encouraging results, it should be noted that the fully decoupling of the motions also brings some disadvantages, such as for example, the increase of the number of joints and a loss of rigidity. It is obvious that it is difficult to solve the problem of the fully decoupling of the motions by keeping the many advantages of the parallel mechanisms. This is why we tried to find a compromise between the decoupling of the motions and the architectural characteristics of the parallel structures.

Another trend of the kinematic decoupling is proposed in the design of the PAMINSA, a new 4-DOF parallel manipulator that has been designed and developed at the I.N.S.A.1 in Rennes. It consists of decoupling the motions in the horizontal plane and the translation along the vertical axis.

Description of the PAMINSA

Fig. 1 shows a 3D model of the PAMINSA with three legs. Each leg of the manipulator is realized by a pantograph mechanism (Fig. 2) with two input points A i and B i , and an output point C i (i = 1, 2, 3). Each input point A i is connected with the rotating drive M i by means of the prismatic guide mounted on the rotating link. This type of manipulator architecture allows the generation of motion in the horizontal plane by the rotating actuators M 1 , M 2 , M 3 and the vertical displacements by the linear actuator M 4 . Thus the displacements (x, y, θ) of the platform in the horizontal plane (xOy) are independent of the vertical displacements z.

In the concept of the PAMINSA, the following properties of the hand operated manipulators (see [START_REF] Arakelian | Equilibrage des manipulateurs manuels[END_REF] are used: the work of the gravitational forces of the manipulated object displaced in the horizontal plane is zero because the gravitational forces are always perpendicular to the displacements. However, the work of the same forces in the case of the vertical displacements is different from zero (the gravitational forces are parallel to the displacements).

Thus, the rotating actuators move the platform in the horizontal plane and the work due to the gravity of the manipulated object is equal to zero. Static properties of PAMINSA were discussed in the study Arakelian et al. (2005), in which was shown that by using an optimal redistribution of masses of the pantograph linkages, the input torques of the rotating actuators of the PAMINSA can be cancelled. In the same study the analytical dynamic modeling of the suggested manipulator by means of Lagrange equations was achieved. Fig. 3 shows the prototype of the PAMINSA, which has been developed at the I.N.S.A. of Rennes. It should be noted that the proposed parallel structure has been patented (Arakelian et al., 2005).

The obvious advantages of PAMINSA

This manipulator has many advantages and it appears to us that the developed manipulator could be used in industrial applications for the manipulation of heavy equipments with great positioning accuracy. We would like to note briefly these advantages: a) a simplification of the vertical control based on the linear input/output equation; b) the improvement of the control in the horizontal plane because the loads of the gravitational forces of the platform on the rotating actuators have been cancelled; c) the improvement of positioning accuracy along the vertical axis because the mechanical locking of the structure does not allow the altitude variations during the displacements in the horizontal plane.

The letter property was discussed in the recent study Arakelian et al. (2005), in which the positioning errors along vertical axis and errors caused by the inclination of the moving platform taking into account the elasticity of links was analyzed. It was shown that the suggested manipulator allows the moving of the platform on the horizontal plane with great accuracy (about 20 µm without charge and 140 µm with a payload of 20 kg applied on the platform). However, PAMINSA has a very good load-carrying capacity and it can support much more. This contribution deals with the methods and new approaches for the improvement of positioning accuracy of PAMINSA.

Improvement of positioning accuracy taking into account the elasticity of links

Many industrial applications of parallel manipulators, such as the assembly of electronic, optical units, or several medical applications require high accuracy. It should be noted that most of parallel manipulators used today are much better at repeatability than at accuracy. For improvement of position accuracy of parallel manipulators, it is possible to use the calibration methods, to increase the rigidity of links or lack of backlashes in drive systems. A new approach called Geometric and Elastic Error Compensation (GEC) was proposed in the study of [START_REF] Meggiolaro | Improving the positioning accuracy of powerful manipulators with application in nuclear maintenance[END_REF]. It was shown that the two techniques can be effectively combined to achieve high absolute positioning accuracy.

In this paper, for improvement of positioning accuracy of PAMINSA, we propose new compensation schemes, which consist of the introduction of complementary units into the initial system making it possible to cancel the positioning errors. Two different approaches are proposed and the performances of such a design are shown.

Improvement of positioning accuracy of PAMINSA by means of correcting systems mounted on the drive system

Most research papers devoted to the study of parallel manipulators deal with the mechanical structures with rigid links and without clearances in the joints. So in this case the position of the platform is considered perfectly parallel to the base. But in reality, the errors due to the elastic deformations of the mechanical structure of the manipulator change the position of the platform (attitude and inclination).

Static rigidity is defined as the 6 6 × symmetrical matrix K that maps generalized

[ ] T z y x z y x M M M F F F = F .
Thus, we have

X K F ∆ ⋅ =
.

(1)

In order to characterize PAMINSA static rigidity, we map the infinitesimal displacements for the whole workspace. The obtained results show that the errors of the elastic deformations of the structure are relatively small. With a payload of 20 kg applied on the platform, the errors, caused by the rotation of the platform about the x and y axes are shown in Fig. 4. The analysis of the obtained results shows that the position in which the structure is less deformed is the central position. When the platform moves away from this position, the manipulator becomes less rigid and loses its accuracy.

These errors are less important if the output point P (Fig. 5), i.e. the end of a surgical device or a sensor, is located on the horizontal plane of the platform. But the error becomes more important if this output point is moved from the horizontal plan of the platform. For example, if the output point is located on the plane xOz and it is moved away 200mm from the horizontal plane of the platform (with 100mm radius), the error 0.14 mm of the platform along the z-axis increases at the end of the output point until 0.57 mm (see Fig. 5). It is obvious that the improvement of positioning accuracy can be achieved by the increase of the rigidity of links. However it is also promising to develop the design methods for improvement of positioning accuracy by additional correcting systems. Fig. 6 shows PAMINSA with two compensation systems, which are presented in Fig. 7. It should be noted that in the modified design of the manipulator the joints on the platform are also changed: the universal joints used in the initial version are replaced by the spherical pairs. The compensation systems, which cancel the errors due to the elasticity of links are provided with two complementary actuators M ci . These actuators allow the displacements of the pantograph's points B i it making possible to eliminate the inclination error of the platform. These modifications allow the correction of the vertical positions of two spherical pairs of the platform, which is sufficient for the cancellation of the positioning error of the inclination of the platform. The vertical positions of such a spherical pair located on the platform can be determined analytically for the whole workspace (or given altitude) of the manipulator on the base of equation (1) or by using three sensors mounted on the platform. The measuring of the spherical joint motion errors for the manipulators with three and six prismatic joints (for Tripod and Hexapod types), as well as several installation examples of the sensors were discussed in the study of [START_REF] Oiwa | Accuracy improvement of parallel kinematic machine[END_REF]. Note that, compensation scheme developed for PAMINSA is constructively more efficient because it is mounted on the drive system of the vertical displacements. As a result, the variable length of the compensation device is always vertical. It should be also noted that it is more simple for computation because the translational displacement in the added system can be found directly from positioning errors of the platform's joint, taking into account the magnification factor of the pantograph linkage. In this manner the significant reduction of errors can be achieved and the obtained results are shown in Fig. 8. It is seen that after compensation of the elasticity errors of links the vertical positions of the platform's joints have constant values and the inclination error is eliminated.

Improvement of positioning accuracy of PAMINSA by means of correcting systems mounted on the platform

Let us consider another correcting system mounted on the platform of PAMINSA. Such a system can be added on any parallel structure for the correction of positioning error of the inclination of the platform. The suggested system consists of a correcting mass, which has the possibility to turn about the vertical axis of the platform and to carry out the translational displacements on the horizontal plane (Fig. 9). Introducing the correcting conditions into equation (1), we obtain

[ ] T y st x st st st M M mg 0 0 0 ) ( ) ( 1 1 - ⋅ + ⋅ = - - K F K ∆X (2)
From which taking into account that after correction In other words the correcting mass m should be located on the platform in such a manner that its gravity eliminated the inclination error of the platform.

For PAMINSA with parameters of the prototype (see Arakelian et al., 2005) In this paper, new design approaches for the improvement of positioning accuracy of 4-DOF parallel manipulator PAMINSA have been discussed. In general, the studies devoted to the parallel manipulators deal with the mechanical structures on the base of rigid body mechanics and consider that the platform is perfectly parallel to the base. Our previous works have shown that the elasticity of links has an influence on the positioning accuracy of the developed parallel manipulator. For cancellation of these positioning errors caused by the elasticity of links, two approaches have been presented. The first solution is realized by means of two correcting systems mounted on the drive system of the vertical displacements. The second solution is carried out by a correcting mass mounted on the platform. The obtained results show that after compensation of the errors caused by the elasticity of links, the vertical positions of the platform's joints have constant values and the inclination of the platform, relating to the base, is cancelled.

Figure 1 .

 1 Figure 1. 3D view of the PAMINSA.Figure 2. Schematics of each leg.

Figure 2 .

 2 Figure 1. 3D view of the PAMINSA.Figure 2. Schematics of each leg.

Figure 3a .

 3a Figure 3a. The prototype of PAMINSA developed at the I.N.S.A. of Rennes.

Figure 3b .

 3b Figure 3b. The CAD model of the prototype.

Figure 4a .

 4a Figure 4a. Errors caused by the rotation of the platform about the x axis.

Figure 4b .

 4b Figure 4b. Errors caused by the rotation of the platform about the y axis.

Figure 5 .

 5 Figure 5. Absolute positioning errors of the output point, which is moved away 200mm from the horizontal plane of the platform.

Figure 6 .

 6 Figure 6. PAMINSA with added compensation systems.

Figure 7 .

 7 Figure 7. Kinematic schema of the added compensation systems for the correction of the positioning errors of the platform along the vertical axis.

Figure 8 .

 8 Figure 8. The vertical positioning errors of the platform's joints C 1 , C 2 and C 3 of the initial and modified manipulators (the examined case correspond to the platform with orientation φ =0° at the altitude z=-0.6 m).

Figure 9 .

 9 Figure 9. Kinematic schema of the correcting system mounted on the platform.

  position λ and orientation α of the correcting mass m .

  the values of the position λ and orientation α for the correcting mass kg m 3 = are shown in Fig. 10.

Figure 10 .

 10 Figure 10. Position λ and orientation α of the correcting mass m .

Institut National des Sciences Appliquées (I.N.S.A.) -National Institute of Applied Sciences.


Sébastien Briot
email: sebastien.briot@ens.insa-rennes.fr

Ilian A Bonev
email: ilian.bonev@etsmtl.ca

ARE PARALLEL ROBOTS MORE ACCURATE THAN SERIAL ROBOTS?

Keywords: parallel robots, performance evaluation, dexterity, error analysis, accuracy ______________________________________________________________________________

It is widely claimed that parallel robots are intrinsically more accurate than serial robots because their errors are averaged instead of added cumulatively, an assertion which has not been properly addressed in the literature. This paper addresses this void by comparing the kinematic accuracy of two pairs of serial-parallel 2-DOF planar robots. Only input errors are considered and all robots are optimized for accuracy, the only constraint being that they cover a given desired workspace. The results of this comparison seem to confirm that parallel robots are less sensitive to input errors than serial robots.

EST-CE QUE LES ROBOTS PARALLELES SONT PLUS PRECIS QUE LES ROBOTS SERIELS ? Résumé

Il est généralement dit que les robots parallèles sont intrinsèquement plus précis que les robots sériels parce que leurs erreurs sont moyennées au lieu d'être ajoutées. Cependant, cette hypothèse n'a jamais été vérifiée dans la littérature. Cet article cherche à répondre à cette question en comparant la précision cinématique de deux paires de robots sériels et parallèles à 2 degrés de liberté. Seules les erreurs sur la commande sont prises en compte et tous les robots sont optimisés afin de les rendre les plus précis possible, la seule contrainte étant qu'ils couvrent tous le même espace de travail. Les résultats de cette comparaison semblent confirmer le fait que les robots parallèles sont moins sensibles aux erreurs sur la commande que les robots sériels.

Mots Clés : robots parallèles, évaluation des performances, dextérité, analyse d'erreurs, précision

Introduction

The development of parallel robots has always been driven by promises of (1) greater rigidity, (2) higher speed, and (3) higher accuracy than serial robots.

The fact that virtually all the hundreds, or even thousands, of motion simulators with load capacities of up to several tons are based on parallel robots (mostly hexapods), with serial robots able to carry at most five hundred kilograms or so, unquestionably demonstrates that the first promise has been fulfilled. The commercial success of the Delta parallel robot and the performance of the recently launched Quickplacer by Fatronik (200 cycles per minute) confirms fulfillment of the second promise, though serial robots are not far behind. But has the third promise been fulfilled yet?

The boom in the development of parallel kinematic machines (PKMs) in the 1990s, particularly those based on hexapods, was driven mainly by that third promise. But none of these hexapods is more accurate than a conventional serial machine tool. Some three-axis and five-axis PKMs are now gaining commercial success, but precision is still not their best feature. While a number of alignment stages are based on parallel robots, the fact remains that great precision is attained by the use of special technologies, such as flexures. Flexures rely on deformation of material to achieve a motion between two elastically joined parts. Flexures are mainly used as passive joints, thus mostly in parallel robots, so it could be said that parallel robots are more accurate for this reason alone. But is it true that parallel robots are kinematically more accurate than serial robots because errors are averaged instead of added cumulatively, as widely claimed:

-"The parallel actuator technology promises to offer […] advantages relative to conventional machine tools, such as […] higher accuracy…" [START_REF] Wavering | Parallel kinematic machine research at NIST: past, present and future[END_REF]; -"Parallel manipulators are preferred to serial manipulators for their […] high positioning accuracy." [START_REF] Rauf | A new measurement device for complete parameter identification of parallel manipulators with partial pose measurements[END_REF]; -"Comparing [sic] to the traditional serial-chain mechanism […], the parallel mechanism exhibits the following advantages: […] better accuracy due to non-cumulative joint error." [START_REF] Song | Error modeling and compensation for parallel kinematic machines[END_REF] ; -"The errors of parallel manipulators are averaged out in the serial chains and the errors of serial manipulator are accumulated [sic]." [START_REF] Guan | Kinematics of a Tricept-like parallel robot[END_REF]; -"Moreover the links [of a serial robot] magnify errors: a small measurement error in the internal sensors of the first one or two links will quickly lead to a large error in the position of the end effector.

[…] The errors of the internal sensors [of a parallel robot] only slightly affect errors on the platform position." [START_REF] Merlet | Parallel Robots[END_REF]. Obviously, the sources of positioning error are numerous (design errors, flexibility of the links, thermal expansion, etc.). But, according to Merlet [START_REF] Merlet | Computing the worst case accuracy of a PKM over a workspace or a trajectory[END_REF], joint sensor errors are the largest source of error in the positioning of a robot. Surprisingly, we have found no reference that explains this "accumulation/averaging of errors" or which compares the input error sensitivity of serial and parallel robots. The most relevant work was reported in [START_REF] Wenger | A comparative study of serial and parallel mechanism topologies for machine tools[END_REF], where several two-degrees-of-freedom (2-DOF) planar serial and parallel robots are compared on the basis of four performance criteria, none of which is purely input error sensitivity (one is called sensitivity, but this takes into account errors in the design parameters, in addition to input errors). Our paper addresses this void and provides some new results regarding the input error sensitivity of serial and parallel robots. In this work, we perform a comparative study of the kinematic accuracy of two serial and two parallel 2-DOF planar robots, one of which was not considered in [START_REF] Wenger | A comparative study of serial and parallel mechanism topologies for machine tools[END_REF]. The only source of error that we consider is that caused by an uncertainty on the input joint sensor measurement (input errors). The robots are compared in pairs, the robots in each pair being subject to identical actuation (and the same input errors). To make the comparison meaningful, all robots are optimized to have the best accuracy, while covering the same desired square workspace area.

In the next section, we will define the four robots and specify the criteria for comparison. In the third section, we will study the maximal position error of each robot, and, in the fourth section, compare the dexterity index to the maximal position error. Conclusions will be presented in the last section.

Criteria for Comparison and Description of the Planar Robots Under Study

In most cases, the so-called dexterity index is used to study the kinematic accuracy of robots [START_REF] Gosselin | The optimum design of robotic manipulators using dexterity indices[END_REF]. Merlet [START_REF] Merlet | Jacobian, manipulability, condition number, and accuracy of parallel robots[END_REF] criticizes such an index, stressing that its major drawbacks are that it mixes both translational and rotational terms of the Jacobian matrix and that it is usually not invariant on the choice of units. As a consequence, the Jacobian matrix must be split into its translational and rotational parts to calculate the dexterity of each of them, but this is not satisfactory for estimating the amplification factor for motion involving both translational and rotational displacements. As we will see in this paper, the dexterity index does not even work properly for robots having only translational degrees of freedom. Thus, the most suitable method for computing the accuracy of robots (actually the input error sensitivity) is to calculate the maximal position error, or orientation error, due to input errors, at a given nominal configuration. This is very easy to do for 2-DOF planar parallel robots using a simple geometrical method. Now that we have decided how to measure the kinematic accuracy of robots, we have to define some criteria for a fair comparison. The first-and most obvious-criterion is that the robots must have the same actuators (only revolute or only prismatic) so that they can have the same input errors. Another criterion is that the robots must be able to accomplish the same task. We impose the constraint in this paper that the robots must be able to displace their end-effectors inside a 1 m by 1 m square (the desired workspace), and do so with the best accuracy possible, meaning that their designs should be optimized to have the smallest mean maximal position error over this desired workspace. This square should obviously be free of singularities. Note that the authors of [START_REF] Wenger | A comparative study of serial and parallel mechanism topologies for machine tools[END_REF] do not compare robots with optimized kinematic accuracy.

That said, we will compare the following two pairs of 2-DOF planar robots for positioning:

a RRRRR parallel robot (Fig. 1a) and a RR serial robot (Fig. 1b); -a PRRRP parallel robot, the directions of its base-mounted prismatic actuators being parallel (Fig. 1c), and a Cartesian serial robot (Fig. 1d).

While the choice of the first pair is fairly obvious, the choice of the second pair might look a bit arbitrary, but it is not. We choose a PP serial robot in which the directions of the prismatic joints are orthogonal, simply because any other non-Cartesian PP serial robot will have worse maximal position error. As for the PRRRP parallel robot, of course, we could choose another architecture with prismatic actuators, but this one is surely the most practical one. Finally, we choose to have the directions of its two prismatic actuators parallel, simply because this gives the most compact design having the desired singularity-free workspace.

Both serial robots are well known and trivial to design to obtain the best accuracy within the desired workspace.

The RR serial robot is designed by finding the optimal values of the parameters OA, AP and d (Fig. 2a). For this robot, it is obvious that the smaller the workspace, the higher the accuracy. Therefore, the design parameters have to define a compact workspace with respect to the desired workspace. The geometric conditions for compactness are:

22 (1 2 ) ( .5 )         OA AP d O , ( 1
)
OA AP d  . ( 2
)
where  is a safety distance added to avoid that the desired workspace includes singularities along its boundary. In this study,  = 0.1 m.

Solving equations ( 1) and (2), we obtain the possible values for OA and AP as functions of d: The mean value of the maximal position error within the desired workspace of the RR serial robots corresponding to any of the two solutions of eq. ( 3) is shown in Fig. 3a, as a function of d. The calculation of the maximal position error will be presented in Section 3. As expected, the optimal design occurs at d = 0 m, and from eq. ( 3), we have OA = 0.67 m and AP = 0.67 m.

( ) / 2 ( ) / 2 OA T d AP T d      or ( ) / 2 ( ) / 2 OA T d AP T d      (3) where 22 (1 2 ) (1/ 2 ) Td        .
The accuracy of the Cartesian serial robot is the same for any position and any actuator stroke. Therefore, there are no optimal design parameters to look for.

The two parallel robots are more difficult to optimize in terms of accuracy. These difficulties are due to the complexity of their direct kinematics and to the presence of singularities inside their workspaces. These two robots have recently been studied in detail [START_REF] Liu | Kinematics, singularity and workspace of planar 5R symmetrical parallel mechanisms[END_REF][START_REF] Liu | Performance atlases and optimum design of planar 5R symmetrical parallel mechanisms[END_REF][START_REF] Liu | On the optimal kinematic design of the PRRRP 2-DOF parallel mechanism[END_REF]. In these references, the authors analyze the robots using different performance indices depending on the link lengths. From [START_REF] Liu | Performance atlases and optimum design of planar 5R symmetrical parallel mechanisms[END_REF], we can roughly estimate that a nearly-optimal RRRRR design occurs when A 1 A 2 = 0.3 m, A 1 B 1 = 0.6 m, and B 1 P = 0.8 m. Although this is not the actual optimal design, for the purposes of our purely qualitative study, it will be good enough. What is important is that the RR serial robot has been given all the chances to win the competition-its design is optimal.

For the PRRRP parallel robot, the shorter the links A i P, the higher the accuracy. While this fact seems obvious, it was nevertheless verified numerically. Thus, it is possible to find a relationship between a and A i P which defines the minimal link length as (Fig. 2b):

( 1) / 2 i A P a     . (4)
The mean value of the maximal position error within the desired workspace of the PRRRP parallel robots whose parameters obey eq. ( 4) is shown in Fig. 3b, as a function of the parameter a. The method for calculating the maximal position error will be presented in Section 3. Figure 3b also shows the required stroke of the actuators (in dashed line). One can see that higher accuracy calls for longer actuators. Thus, we chose a nearly-optimal design at A 1 P = A 2 P = 2.1 m and a = 3 m.

Figure 4 shows the optimized designs of three of the robots under study, their workspaces and the square within them that constitutes the desired workspace, all to the same scale. The workspace of the Cartesian serial robot, which is not shown here, is obviously a rectangular region. In the next section, we will analyze the maximal position errors of these four robots using a geometrical method.

Analysis of the Maximal Position Error of the Robots Under Study

Comparison of the RRRRR parallel robot and the RR serial robot

For these robots, we consider that the maximal input error is equal to ±210 -4 rad. The maximal position error for these robots is quite easy to determine. For each of them, this error occurs at one of the four sets of extreme input errors, i.e., at one of the corners of the so-called uncertainty zone, as shown in Fig. 5.

Thus, it is possible to calculate the maximal position error at each nominal position. These errors are presented in Fig. 6. In addition, Table I gives some statistics regarding the maximal position error for each robot over the desired workspace.

Comparison of the PRRRP parallel robot and the Cartesian serial robot

For these robots, we consider that the maximal input error is equal to ±100 µm. The maximal position error for these robots is also quite easy to determine. It is equal to approximately 141 µm for the Cartesian serial robot (Fig. 7b). For the PRRRP parallel robot, this error occurs at one of the four sets of extreme input errors, i.e., at one of the corners of the so-called uncertainty zone, as shown in Fig. 7a.

Thus, it is possible to obtain the maximal position error at each position for the PRRRP parallel robot. This maximal position error is virtually equal to the input error, i.e., ±100 µm, for any position. Therefore, no contour plot as in Fig. 6 is given for this robot. Table II gives statistics regarding the maximal position error for each robot over the desired workspace. In concluding this section, we warn readers that our study is too limited to draw any general conclusions. It is quite possible, for example, that if the desired workspace is different (e.g., an annular region or an elongated rectangular region) or if the input errors are much smaller, some of the results could be quite different.

Nevertheless, our study suggests that a RRRRR parallel robot is much less sensitive to input errors than an equivalent RR serial robot, while having nearly the same overall dimensions. (We again point out that by "equivalent" we mean that the robots have the same desired square workspace and the same input errors.) Similarly, a PRRRP parallel robot is much less sensitive to input errors than an equivalent Cartesian serial robot, but only when its overall dimensions are much greater than those of the serial robot. In fact, the mean maximal position error of a nearlyoptimal PRRRP parallel robot is equal to its maximal input error, which means that both the Cartesian robot and the parallel robot are dimension invariant; hence the comparison is fair. However, with greater dimensions there are more manufacturing errors, and a need for larger actuators stroke, which means higher manufacturing costs. For example, the actuators of the PRRRP parallel robot should be three times as long as those of the Cartesian robot. Thus, we find it hard to believe that, in practice, a PRRRP parallel robot would be more precise than a Cartesian robot.

Finally, we are tempted to comment on this widely claimed "accumulation of errors" in serial robots in contrast to the "averaging of errors" in parallel robots. A ±100 µm input error produces a maximal position error of approximately 141 µm in a Cartesian serial robot, and a maximal mean positioning error of about 100 µm in a (optimally designed) PRRRP parallel robot. So, in this example, one might indeed say that there is an averaging of errors in the parallel robot. However, when its end-effector is close to certain singularities, then the maximal position error could be several times larger than the input error (in the case of prismatic actuators). We therefore believe that this is, in general, too strong a statement and should be avoided.

Analysis of the Dexterity Index

In this section, we will compare the dexterity indices of the four robots under study to their maximal position errors over the desired workspace. The dexterity of a robot is calculated as defined in [START_REF] Merlet | Jacobian, manipulability, condition number, and accuracy of parallel robots[END_REF]:

1 1   JJ  , (5)
where we use the Euclidean norm defined as

1 2 T tr     J JJ , ( 6
)
and J is the Jacobian matrix of the robot. The Jacobian matrices for the four robots under study will not be derived here, since this is fairly simple to do and the calculation of the dexterity index is not the main subject of this paper.

The dexterity maps for the two parallel robots and for the RR serial robot are represented in Fig. 8. The dexterity index of the Cartesian serial robot is constant and equal to unity, and is therefore not shown in this figure .  Several observations can be made. Firstly, the shape of the dexterity map does not correspond to the shape of the maximal position error map. Areas of highest dexterity do not correspond to areas of lowest maximal position error. Secondly, the values of the dexterity indices of the RRRRR parallel robot and of the RR serial robots are similar, even though the first robot is more accurate than the second in terms of maximal position error. Likewise, the values of the dexterity index for the PRRRP parallel robot are not constant, even though its maximal position error is nearly constant.

It is therefore clear that the dexterity map cannot be used to evaluate the global nature of the accuracy of a robot, let alone to compare the accuracy of different robot designs, and this is true even for robots having only positioning capabilities (i.e., when there are no problems involving mixed units).

Conclusions

Obviously, our study is quite simplistic and too limited for us to draw any general conclusions as numerous other authors have done who claim that parallel robots are more accurate than serial robots. Yet, in spite of the limited nature of our study, it still suggests that parallel robots are indeed theoretically more accurate than serial robots, when input errors are assumed to be the only source of inaccuracy. This was shown using the natural concept of maximal position error instead of the dexterity index, which proved to be meaningless for comparing accuracy.

But, are parallel robots more accurate than serial robots in practice, where the larger dimensions required in parallel robots might induce greater mechanical errors? Moreover, are we not, after all, comparing apples and oranges when we state that parallel robots are more accurate than serial robots? How can we say, for example, that a hexapod (with six linear actuators) is theoretically more accurate than a serial machine tool (with three linear and three rotary actuators)? The truth is that only practice, and not theory, will show whether or not parallel robots can be manufactured to be more accurate than serial robots. In other words, we believe that the mechanical design of a robot, its manufacture and its calibration are much more important drivers of accuracy than the optimal kinematic design. So, the question is whether parallel robots can be mechanically designed, manufactured and calibrated so as to be more precise than the most accurate serial robots on the market. For the time being, we can only claim that parallel robots seem to be less sensitive to input errors than serial robots.

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3

 3

( a )Fig. 1 . 5 (Fig. 2 .

 a152 Fig. 1. The two pairs of planar robots under comparison (not to scale).

______________________________________________________________________________ 6 (Fig. 3 .

 63 Fig. 3. Variations of the mean value of the maximal position error over the desired workspace.

( a )Fig. 4 .

 a4 Fig. 4. Workspace of the planar robots under study (to scale).

______________________________________________________________________________ 7 (Fig. 5 .Fig. 6 .

 756 Fig. 5. Uncertainty zones for the first two planar robots.

( a )Fig. 7 .

 a7 Fig. 7. Uncertainty zones for the second pair of planar robots.

Fig. 8 .

 8 Fig. 8. Contour plots of the dexterity index for three of the robots over the desired workspace.

TABLE I

 I

	Statistics for the Maximal Position Errors
	Max. position error	RRRRR parallel robot RR serial robot
	Max. value (µm)	197	367
	Mean value (µm)	175	240
	Standard deviation (µm) 24	61

TABLE II

 II

	Statistics for the Maximal Position Errors
	Max. position error	PRRRP parallel robot PP serial robot
	Max. value (µm)	100	141
	Mean value (µm)	100	141
	Standard deviation (µm) 0	0