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

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

Victor Glazunov

Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure

Keywords: Parallel manipulator, singularity-free zones, pressure angle, force transmissivity analysis

This paper is focused on the study of singularity of planar parallel manipulators taking into account the force transmission, i.e. study of singularity of planar manipulator by introducing the force transmission factor. Thus the singularity zones in the workspace of the manipulator are defined not only by kinematic criterions from the theoretical perfect model of the manipulator but also by the quality of force transmission. For this purpose, the pressure angle is used as an indicator of force transmission. The optimal control of the pressure angle for a given trajectory of the manipulator is realized by means of legs with variable structure. The suggested procedure to determination of the optimal structure of the planar parallel manipulator 3-RPR is illustrated by two numerical simulations.

Introduction

It is well-known that the closed-loop of parallel manipulators limits the motion of the platform and creates special singular zones inside the workspace [START_REF] Merlet | Parallel robots[END_REF]. The workspace of the parallel manipulators which is less than the serial manipulators becomes smaller and limits their functional performance.

One of the most evident solutions of this problem is the introduction in the initial system of complementary actuators, which make it possible to eliminate the singular configurations of the parallel manipulator by means of optimal control of the motion [START_REF] Glazunov | Parallel manipulator control while intersecting singular zones[END_REF][START_REF] Alvan | On the control of the spatial parallel manipulators with several degrees of freedom[END_REF]. However, it is an expensive solution to the problem because of the additional actuators and the complicated control of the manipulator caused by actuation redundancy.

In this paper we propose a new solution, which carried out by using mechanisms of variable structure, i.e. a mechanism whose structure parameters can be altered. With regard to the determination of singularity-free zones inside the workspace we propose a kinetostatic approach taking account of the force transmission.

The physical interpretation of a singularity in kinematics refers to those configurations in which the number of degrees of freedom of the mechanical structure changes instantaneously, either the manipulator gains some additional, uncontrollable degrees of freedom or loses some degrees of freedom. Algebraically, a singularity analysis is based on the properties of the Jacobian matrices of the mechanical structure, i.e. when the Jacobian matrices relating the input speeds and the output speeds become rank deficient [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Ma | Architecture singularities of parallel manipulators[END_REF][START_REF] Zlatanov | Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics[END_REF][START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF][START_REF] St-Onge | Singularity analysis and representation of the general Gough-Stewart platform[END_REF][START_REF] Pernkopf | Singularity analysis of spatial Stewart-Gough platforms with planar base and platform[END_REF][START_REF] Wen | Singularities in three-legged platform-type parallel mechanisms[END_REF][START_REF] Bandyopadhyay | Analysis of configuration space singularities of closedloop mechanisms and parallel manipulators[END_REF][START_REF] Zhao | Analysis of the singularity of spatial parallel manipulator with terminal constraints[END_REF][START_REF] Ben-Horin | Singularity analysis of a class of parallel robots based on Grassmann-Cayley algebra[END_REF][START_REF] Li | Determination of maximal singularity-free zones in the workspace of planar three-degree-of-freedom parallel mechanisms[END_REF][START_REF] Glazunov | Classification principles and analysis methods for parallel-structure spatial mechanisms[END_REF][START_REF] Glazunov | Twists of movements of parallel mechanisms inside their singularities[END_REF]. However, it is also wellknown that when the parallel manipulator is close to a singular configuration, it loses the stiffness and the quality of motion transmission, as a result, the payload capability. Thus, the singularity zones must be avoided and an indicator of the quality of motion transmission close to the singular configurations of parallel manipulators must be defined. In the present work we use a kinetostatic approach for the evaluation of the quality of motion transmission by using the pressure angle, well-known in the mechanism design but not so often applied to the parallel mechanisms. One defines the pressure angle as an angle between vectors of force and velocity of a point at which the force is applied. Thus for best force transmission it would be desirable if the pressure angle will be close to zero. One also knows the transmission angle, which is 90° minus pressure angle and accordingly should be desirable if it will be close to 90°.

S. Balli and S. Chand [START_REF] Balli | Transmission angle in mechanisms[END_REF] considered several examples for determination of transmission angle of planar and spatial mechanisms, particularly, for mechanisms with two degrees of freedom. G. Sutherland and B. Roth [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF] showed that the input link of a spatial mechanism tends to move the output link when the transmission wrench is not reciprocal to the output link twist. On the base of this consideration a general index of motion transmission for spatial mechanisms is proposed. The quality of motion and force transmission was successfully summarized in the work of G. Sutherland [START_REF] Sutherland | Quality of motion and force transmission[END_REF] and C.-C. Lin, W.-T. Chang [START_REF] Lin | The force transmissivity index of planar linkage mechanisms[END_REF]. The study of G. Sutherland and B. Roth [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF] was generalized for any spatial single-loop mechanism in the recent study C. Chen and J. Angeles [START_REF] Chen | A generalized transmission index for spatial linkages[END_REF]. O. Alba-Gomez, P. Wenger and A. Pamanes [START_REF] Alba-Gomez | Consistent kinetostatic indices for planar 3-DOF parallel manipulators, application to the optimal kinematic inversion[END_REF] have evaluated the quality of motion in the three degrees of freedom manipulators by means of a kinetostatic indicator, which is similar to the pressure angle. Among several works may be distinguished also the study of J. Lee, J. Duffy and J. Keler, J. [START_REF] Lee | The Optimum Quality Index for the Stability of In Parallel Planar Platform Devices[END_REF].

The singularity determined from classical approach taking into account only kinematic aspects give information about some singular positions in the geometrical sense. However, in this case, there is not any information about the zones close to these positions, in which the manipulator loses the quality of motion. In this paper the pressure angle are used for determination of these zones close to the singular positions, which cannot be reached by manipulator.

The singularity analysis can be executed on the base of velocity equations. The left hand side of these equations represents the reciprocal screw products of wrenches acting on the moving platform from kinematic chains and the twist of motion of the platform. The right hand side represents the reciprocal screw products of the same wrenches and the twists corresponding to the actuated kinematic pairs. The singularity of type one exists if all the twists of one of the kinematic chains are linear dependent. The singularity of type two exists if the wrenches acting on the platform from the kinematic chains are linear dependent. These singularities can be detected if the determinants consisting of the Plücker coordinates of mentioned twists (the type one singularity) or of mentioned wrenches (the type two singularity) are equal to zero [START_REF] Glazunov | Classification principles and analysis methods for parallel-structure spatial mechanisms[END_REF]. But from the point of view of force transmission invalid configuration can appear even when the determinant consisting of the Plücker coordinates of the wrenches acting on the moving platform from the kinematic chains is not equal to zero. That is why we consider m pressure angles (where m is the degree of freedom) and choose the worst of these angles as the criteria of closeness to singularity.

In the present study, we use the pressure angle as an indicator of the quality of motion transmission, and in our opinion such a kinetostatic approach shows the nature of the inaccessibility of parallel manipulators' singular zones better than the kinematic approach.

The quality of motion transmission and pressure angle

Let us consider a planar parallel manipulator (Figure 1), which consists of the base, the output link (the moving platform) and three kinematic chains with two revolute pairs . Thus, such a manipulator with three actuators (rotating or linear) has three degrees of freedom. The moving platform can translate in the xy plane and rotate (angle  ) with respect to an axis perpendicular to the xy plane. The workspace of the manipulator can be defined as the totality of positions that a moving platform can reach. However, these accessible positions are limited not only by geometrical parameters and the type of actuation of the parallel mechanism but also by force transmission. Especially in the configurations close to the singular positions the force transmission becomes unfavorable and the transmission of motion can be disrupted, as a result, leads to a breakdown of the parallel mechanism or an undesirable motion. The pressure angle is an important criterion for the analysis of the inaccessible zone of parallel manipulators.

However, when the number of links and number of degrees of freedom increase, the determination of the pressure angles becomes more complicated. Let us examine the pressure angles of the considered manipulator. Let's consider that the revolute pairs Thus, each kinematic chain includes one actuated and two passive pairs. The wrench acting to the output link is reciprocal to the unit vectors situated along the axes of nonactuated pairs. Let E i1 , E i2 , E i3 (Figure 2) be the unit vectors of the axes of kinematic pairs, where i (i = 1, 2, 3) is the number of the chain.

Here E i1 corresponds to rotating actuated pair, E i2 and E i3 correspond to sliding and rotating passive pairs accordingly (E i1 and E i3 directed perpendicular to the plane of the mechanism). These unit screws in any position of the mechanism have Plücker coordinates:

  0 1 0 0 0 1 0 1 1 y i x i i e e E ,   0 0 0 0 0 2 0 2 2 y i x i i e e E ,   0 1 0 0 0 3 0 3 3 y i x i i e e E ,
where

Ai x i y e  0 1 , Ai y i x e   0 1 , i Ai Bi x i l x x e / ) ( 0 2   , i Ai Bi y i l y y e / ) ( 0 2   , Bi x i y e  0 3 , Bi y i x e   0 3 , x Ai , x Bi , y Ai , y
Bi are the coordinates of the point A i and B i , l i is the distance between the points A i and B i (i = 1, 2, 3). For planar mechanisms 3-entries screws can be used [START_REF] Dimentberg | The screw calculus and its applications in mechanics[END_REF]. The Plücker coordinates of the unit screws can be described in the matrix (E) i (i = 1, 2, 3):

                 y i z i y i z i y i z i i e e e e

(E)

The determinant of the matrix (E) vanishes if the axes E i1 and E i3 coincide. It means the occurrence of singularity when the actuator causes only rotation in the joint E i3 .

We can obtain the wrenches R i (i = 1, 2, 3), which are reciprocal to the unit vectors of the axes of the passive kinematic pairs [START_REF] Dimentberg | The screw calculus and its applications in mechanics[END_REF]. They can be written as:

  0 0 0 0 iz iy ix i r r r R (i = 1, 2, 3
). The conditions of reciprocity are:

0 0 2 0 2   iy y i ix x i r e r e ; 0 0 0 3 0 3    iz iy y i ix x i r r e r e ( 1
)
The equation ( 1) means that each connecting kinematic chain determines one wrench of zero pitch (vector). It is perpendicular to the axis E i2 and intersects the point B i . The coordinates of wrenches in the form of the matrix (R) is given by:

              z y x z y x z y x r r r r r r r r r 3 3 3 2 2 2 1 1 1 (R)
In singular configurations the system of the wrenches R i degenerates and they intersect in the same point or are parallel [START_REF] Bonev | Singularity analysis of 3-DOF planar parallel mechanisms via Screw Theory[END_REF]. It can be shown by the representation of the components of this matrix. If all the wrenches are parallel then the first two columns are proportional. If all the wrenches intersect in the same point

  0 W W y x W
then the coordinate 0 iz r (i = 1, 2, 3) can be written as

W y W x iz x r y r r 1 1 0  
. In this case in the matrix (R) the third column is a linear combination of the first and seconds columns.

To find the pressure angle we consider the wrenches R i and the directions of the velocities of the points B i determined by the twists reciprocal to these wrenches. The velocity of the point B 1 is determined by two wrenches R 2 and R 3 . One can find the twist

  0 0 0 1 1 1 1 y x z v v w W
reciprocal to the wrenches R 2 and R 3 using the equations:

0 0 2 1 2 1 2 1    z z y y x x r w r v r v ; 0 0 3 1 3 1 3 1    z z y y x x r w r v r v (2)
It is obvious that the axis of the twist W 1 is situated perpendicular to the plane of the mechanism and intersects the center Q 1 of velocities of the platform according to the wrenches R 2 and R 3 (Figure 2). Without interruption of generality the twist W 1 can be expressed as  

0 1 0 0 1 1 Q Q x y 
. The velocity V B1 has the coordinates

1 1 1 1 1 1 B Q B z x xB y y y w v v     , 1 1 1 1 1 1 B Q B z y yB x x x w v v     
. If the wrenches R 2 and R 3 are parallel (

x x r r 3 2  , y y r r 3 2  , 0 3 0 2 z z r r  ) then 0 1  z w
and V B1 is perpendicular to these wrenches R 2 and R 3 . Finally, the pressure angle can be written as:

  1 1 1 1 1 / arccos R V R V B B   (3)
It was noted that in the singular configurations all the pressure angles are equal to 90°. Indeed, in this case the axis of the wrench R 1 intersects the axes of the wrenches R 2 and R 3 and the velocity V B1 is perpendicular to the axis of the wrench R 1 .

Thus, the pressure angles can be determined at the joints of each kinematic chain then the maximum value of the pressure angles can be compared with their limit values. In this way we have mapped whole workspace of the parallel manipulator to detect the inaccessible zones with unfavourable values of the pressure angles.

It should be noted that the singularity analysis can be executed on the base of velocity equations. The left hand side of these equations represents the reciprocal screw products of wrenches acting on the moving platform from kinematic chains and the twist of motion of the platform. The right hand side represents the reciprocal screw products of the same wrenches and the twists corresponding to the actuated kinematic pairs. The singularity of type one exists if all the twists of one of the kinematic chains are linear dependent. The singularity of type two exists if the wrenches acting on the platform from the kinematic chains are linear dependent. These singularities can be detected if the determinants consisting of the Plücker coordinates of mentioned twists (the type one singularity) or of mentioned wrenches (the type two singularity) are equal to zero [START_REF] Glazunov | Classification principles and analysis methods for parallel-structure spatial mechanisms[END_REF]. But from the point of view of force transmission invalid configuration can appear even when the determinant consisting of the Plücker coordinates of the wrenches acting on the moving platform from the kinematic chains is not equal to zero. That is why we consider m pressure angles (where m is the degree of freedom) and choose the worst of these angles as the criteria of closeness to singularity. Thus the standard screw method allows the determination of singular positions only in geometrical sense. By using the pressure angle we determine the volumes, in which the force transmission ability is invalid.

If the prescribed path of the parallel manipulator intersects any unacceptable zone in which the pressure angle has an inadmissible value the transmission of the motion can be disrupted. In this case, it is necessary to change the structural parameters of the mechanism, i.e. the input motions. It will be shown in the following section. By example, one or all of the pairs A i (Figure 1) can be passive and the prismatic pairs (i = 1, 2, 3) can be actuated by the chain A i D i C i . In this case, the actuator torque is transmitted to the link A i D i , which becomes an input link and moves the prismatic pair. It should be noted that the mobility of the modified manipulator is not changed and it is always equal to three.

Legs with variable structure

Let us consider the system of wrenches existing in this case. In the previous case by fixed actuator the link B i C i had translation mobility along the axis of the prismatic pair. In this case the link B i C i is constrained by two wrenches of zero pitch T i1 and T i2 . The wrench T i1 is reciprocal to the unit screws of the axes of two kinematic pairs. One of them is rotating and its axis intersects the plane of the mechanism in the point A i . The second of them is prismatic and its axis is directed along the line A i B i . The Plücker coordinates of the unit screws of the axes of these kinematic pairs are the same as the coordinates of

  0 0 0 0 iz iy ix i r r r R (i = 1, 2, 3
) can be determined using the equation analogous to [START_REF] Merlet | Parallel robots[END_REF]. The pressure angle can be found using the equation (3).

Thus, in each position we determine m pressure angles corresponding to all m degrees of freedom. Then we consider the maximum value of these angles. By such a way, we can determine the pressure angles corresponding to the different structures distinguished by different input links and obtain all possible workspace with singularity-free zones. It is examined in the next section.

Numerical plotting of singularity-free zones taking account of pressure angle

In this section we would like to show the singularity-free zones in the workspace of 3-RPR parallel manipulator with modified legs. These zones have been determined by using the maximum values of the pressure angles.

For numerical simulation we consider 3-RPR parallel manipulator in which the basic triangle

3 2 1 A A A
is equilateral with radius 0.35 m (Fig. 1) and the platform also represents an equilateral triangle with radius 0.1 m. In other words the centers of the joints mounted on the frame are disposed on the circle with radius 0.1m. The rotation of the revolute joints

  m C B C A i i i i 05 . 0 ) ( min min   .
Taking into account that the manipulator can be actuated either by links

  1 q  RRP : RPR-RPR-RPR : T 3 2 ) 1 ( ) , , ( 
RPR : RPR-RPR-RPR :

T 3 2 ) 3 ( ) , , (    1 q  RPP : RPR-RPR-RPR : T 3 2 ) 4 ( ) , , (    1 q  PRR : RPR-RPR-RPR : T 3 2 ) 5 ( ) , , (   1 q p  PRP : RPR-RPR-RPR : T 3 2 ) 6 ( ) , , (    1 q  PPR : RPR-RPR-RPR : T 3 2 ) 7 ( ) , , (   1 q p  PPP : RPR-RPR-RPR : T 3 2 ) 8 ( ) , , (   1 q p 
Tables 1 and2 show the workspaces of each case of actuation with 0° and 45° orientation angles (the origin of the fixed base frame is located at the centre of the equilateral triangle ). In these figures, several zones can be seen, which correspond to the variations of the maximum values of the pressure angle for given position of the platform. The contrast intensity shows the variations of the pressure angle (see Figure 4). Thus, the black zones are the surfaces where the pressure angle has inadmissible values, and as a result, these are the zones, which cannot be reached by the parallel mechanism. The table 3 shows the ratio between the total value of singularity-free volumes and the total workspace for each case of actuation (for two examined cases: Figure 5 shows the reachable workspace of the modified parallel mechanism with legs of variable structure. We can see that the workspace of the modified manipulator is only composed of singularity-free zones and the whole workspace of the manipulator is reachable (increase until 100%).

  0  and   45  ).

Trajectory planning and design procedure for determination of optimal leg structure

In order to obtain the best structural architecture of the manipulator for a given trajectory, in this section we describe a procedure, which allows determining the optimal system of actuation. This algorithm is based on the control of the pressure angles in the joints of the manipulator along the given trajectory (Fig. 6). Fig. 6. Procedure to determination of the optimal structure of the parallel manipulator taking into account the limit pressure angle.

Two numerical examples are considered below in order to illustrate the application of the suggested design procedure.

Problem No. 1. For given parallel manipulator (Fig. 1) with legs of variable structure (Fig. 3) generate the trajectory by straight line from the initial position ) 0 , 0 , 0 ( In order to illustrate the variations of torques for examined case we develop a model of the manipulator with the given trajectory using the ADAMS software. A force parallel to the x-axis and equal to 100 N was applied to the platform and the friction coefficients in the prismatic pairs were equal to 0.01. The obtained torques are shown in Figure 7. We can note that the torques have admissible values along the trajectory.

YES

Input data: the geometrical parameters of the parallel mechanism, the given trajectory and the limit value of the pressure angle Estimation of the pressure angles in the joints along the trajectory for all possible structures of the parallel mechanism with variable architecture (pressure angles are less than the limit value)

This parallel manipulator cannot carry out the given trajectory

The possibility of the motion generation by one structure for which the maximum value of the pressure angle along the trajectory is always less than the limit value Problem No. 2. For given parallel manipulator (Fig. 1) with legs of variable structure (Fig. 3) generate the trajectory by straight lines from the initial position ) 0 , 0 , 0 ( . In this case, the estimation of pressure angle shows that it is impossible to carry out the given trajectory by one structural system. In a first time, the trajectory from initial position ) 0 , 0 , 0 ( It should be noted that the mechanism of variable structure shown above was developed by means of the added articulated dyads, but, it is obvious that such a mechanism can be designed on the base of the screw or cam systems, the rhombic pantographs, etc.

In a similar way one obtains the increase of singularity-free zones in the workspace of parallel manipulators only with revolute pares. In Fig. 9 is illustrated a 3-RRR parallel manipulator with the legs of variable structure. A (i = 1, 2, 3). In this way we can obtain the leg's mechanisms with different structural parameters and carry out the given trajectory taking into account the limit value of the pressure angle. We shall not treat the procedure of resolution that it differs from the previous case only by determination of the pressure angle.

Conclusions

A procedure for the increase of singularity-free zones in the workspace of planar parallel manipulators has been presented in this paper. The procedure is based on the known kinematic singularity equations and the control of the pressure angles in the joints of the manipulator along the given trajectory of the platform. The zones, which cannot be reached by the manipulator, were detected. For increase of the reachable workspace of the manipulator the legs of variable structure were proposed. Such a solution allows obtaining the best structural architecture of the manipulator for any trajectory. The design of the optimal structure of the planar parallel manipulator 3-RPR was illustrated by two numerical simulations. We believe that the suggested method is a useful tool for the improvement of the functional performance of parallel manipulators with singular zones.

Finally, it should be noted that the same problem for spatial parallel manipulators was studied in [START_REF] Arakelian | Improvement of Functional Performance of Spatial Parallel Manipulators Using Mechanisms of Variable Structure[END_REF].

Fig. 1 .

 1 Fig. 1. Planar parallel manipulator 3-RPR.

Fig. 2 .

 2 Fig. 2. Representation of the planar parallel manipulator 3-RPR in 3D.

Figure 3 Fig. 3 .

 33 Figure3shows a schematic of the modified leg with the added articulated dyad. The rotating actuators are mounted on the base and connected by electromagnetic clutches with

  to the orientation of links A i D i with the mechanism in the central symmetric configuration. For added dyads m dyads are always located on the left of the prismatic pairs as it is shown in Figure3and the translation of the prismatic pairs are limited relative to the joints

  platform, we have 8 different combinations of actuation, i.e. we have 8 different combinations of input parameters presented below (underlined letters show the input pairs, "R" for input links

Fig. 4 .

 4 Fig. 4. The contrast intensity corresponding to the pressure angle.

  Actuators: RRP (h) Actuators: PPR

Fig. 5 .

 5 Fig. 5. The reachable workspace of the parallel manipulator with modified legs.

  the pressure angle along the given trajectory shows that the best structural solution for generation of motion is the RPR-RPR-RPR mechanism, i.e. when the first actuator is connected with the link the maximum values of the pressure angles in the joints are always less than the limit value.

Fig. 7 .

 7 Fig. 7. Torques of the actuators.

  out by the RPR-RPR-RPR mechanism then from the second position ) -RPR-RPR mechanism. Thus the suggested solution based on these structural architectures allows obtaining the optimal actuation system of the manipulator considering the pressure angle.The obtained torques are shown in Figure8. We can note that the torques have admissible values along the trajectory but there is a discontinuity in the point 2 P caused by the structural change of the parallel mechanism.

Fig. 8 .

 8 Fig. 8. Torques of the actuators.

Fig. 9 .

 9 Fig. 9. Planar parallel manipulator 3-RRR with legs of variable structure.

  two input links cannot actuated simultaneously and the input motion can be transmitted either by the link

Table 3

 3 Total value of singularity-free volumes for each case of actuation

	Type of	= 0 deg (workspace surface: 0.21 m²)  = 45 deg (workspace surface: 0.2 m²)
	actuatio		Singularity-free		Singularity-free
	n	Singularity-free	zones relative to	Singularity-free	zones relative to the
		zones (m²)	the whole	zones (m²)	whole workspace
			workspace
	RRR	0.137	65%	0.147	74%
	PPP	0.181	86%	0.152	76%
	PRR	0.152	72%	0.158	79%
	RPR	0.152	72%	0.158	79%
	RRP	0.152	72%	0.158	79%
	RPP	0.155	74%	0.165	83%
	PRP	0.155	74%	0.165	83%
	PPR	0.155	74%	0.165	83%

Acknowledgement

The authors thank Philippe Wenger and Damien Chablat (IRCCyN of Nantes) for very useful discussions while editing of the manuscript.


This paper deals with the complete shaking force and shaking moment balancing of the position-orientation decoupled PAMINSA manipulator. The dynamic reaction forces on the manipulator's base are eliminated by making the total mass center of the moving links stationary. The reaction moments on the frame are eliminated by optimal control of the end-effector, which rotates with prescribed velocity. The numerical simulations carried out using ADAMS software demonstrate that the balanced manipulators transmit no inertia loads to their bases.

I. INTRODUCTION

HE problem of mass balancing of mechanisms is of continuing interest to researchers and different approaches and solutions have been developed and documented [START_REF] Lowen | Balancing of linkagesan update[END_REF][START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF][START_REF] Arakelian | A historical review of the evolution of the theory on balancing of mechanisms[END_REF]. But despite its long history, mechanism balancing theory continues to develop and new approaches and solutions are constantly being reported. A new field for their application is the design of mechanical systems for fast manipulation, which are very efficient for advanced robotic applications. The cancellation of the shaking force transmitted to the robot frame can be achieved by a traditional approach, which consists of the fixation of the total mass center of the manipulator moving links [START_REF] Agrawal | Reactionless space and ground robots: novel designs and concept studies[END_REF][START_REF] Wang | Static balancing of spatial three-degreeof-freedom parallel mechanisms[END_REF][START_REF] Newman | The optimal control of balanced manipulators[END_REF][START_REF] Laliberté | Static balancing of 3-DOF planar parallel mechanisms[END_REF][START_REF] Fujikoshi | Balancing apparatus for jointed robot[END_REF][START_REF] Wang | Static balancing of spatial four-degreeof-freedom parallel mechanisms[END_REF][START_REF] Russo | Static balancing of parallel robots[END_REF]. However, the cancellation of the shaking moment is a more complicated task. The literature review of works on shaking moment balancing shows that the methods can be classified into three principal groups: (i) shaking moment balancing using counter-rotations [START_REF] Berkof | Complete fore and moment balancing of inline four-bar linkages[END_REF][START_REF] Arakelian | Complete shaking force and shaking moment balancing of linkages[END_REF][START_REF] Dresig | Vollständiger und harmonischer Ausgleich ebener Mechanismen[END_REF][START_REF] Herder | A counter-rotary counterweight for light-weight dynamic balancing[END_REF][START_REF] Kochev | General theory of complete shaking moment balancing of planar linkages: a critical review[END_REF], (ii) shaking moment balancing by adding four bar linkages [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF][START_REF] Wu | On the synthesis of a reactionless 6-DOF parallel mechanism using planar four-bar linkages[END_REF][START_REF] Gosselin | Synthesis and design of reactionless three-degree-of-freedom parallel mechanisms[END_REF][START_REF] Foucault | Synthesis, design, and prototyping of a planar three degrees-of-freedom reactionless parallel mechanism[END_REF][START_REF] Wu | Design of reactionless 3-DOF and 6-DOF parallel manipulators using parallelepiped mechanisms[END_REF] and (iii) shaking moment balancing by optimal trajectory planning [START_REF] Papadopoulos | Design and motion planning for a zero-reaction manipulator[END_REF][START_REF] Fattah | On the design of reactionless 3-DOF planar parallel mechanisms[END_REF][START_REF] Arakelian | Dynamic Balancing of the Scara Robot[END_REF].

The aim of the first approach is to connect the counterweights with planetary gear trains, which generate counter-rotations. The disadvantage of such a balancing is the need for connection of gears to the oscillating links. The oscillations of the links of the mechanism will create noise unless expensive anti-backlash gears are used.

The second approach is achieved by building a mechanical system out of modules -four-bar linkages or parallelepiped mechanisms. The linkages with prescribed geometric parameters allow the optimum redistribution of Manuscript received January 31, 2009. S. Briot is with the Institut de Recherche en Communication et Cybernétique de Nantes (IRCCyN), Nantes, 44321 France. V. Arakelian is with the National Institute of Applied Sciences (I.N.S.A.), Rennes, 35043 France (corresponding author: +33223238492; fax: +33223238726; e-mail: vigen.arakelyan@insa-rennes.fr).

the moving mass to achieve shaking moment balancing. The originality of such a solution is the balancing of the shaking moment without counter-rotations. However, balancing by using this method is only attained via a considerably complicated design.

In the third approach the shaking moment is balanced through optimum motion planning. In study [START_REF] Papadopoulos | Design and motion planning for a zero-reaction manipulator[END_REF], a 3-DOFs manipulator is developed in order to carry out an in-plan positioning task in which a redundant degree of freedom is used for shaking moment balancing. The drawback of this approach consists of complementary actuation. It should be noted that the shaking moment of a force balanced 2-DOFs manipulator can be successfully cancelled by a complementary actuator mounted on the frame and rotated with prescribed velocity. Another approach based on the optimum motion planning has been developed for the 3-RRR parallel manipulator [START_REF] Fattah | On the design of reactionless 3-DOF planar parallel mechanisms[END_REF], in which the trajectory in the horizontal plane of the manipulator is imposed. The rotation of the platform is calculated in such a manner that the angular momentum of the parallel mechanism becomes constant. It should be noted that the legs of a force balanced manipulator have important masses and consequently inertia moment. The rotation of the platform is used for compensation of their moments. However, the platform of a 3-RRR parallel manipulator cannot perform a full-cycle rotation, and as a result the rotary acceleration is limited. In our previous study [START_REF] Arakelian | Dynamic Balancing of the Scara Robot[END_REF] the complete shaking force and shaking moment of the SCARA robot with 4-DOF has been discussed. Reaction moments on the frame of the SCARA robot are eliminated by optimal control of the end-effector, which rotates with prescribed velocity. In the present paper such an approach is used for the balancing of the positionorientation decoupled PAMINSA manipulator.

The paper is organized as follows. At first the concept of the manipulator is discussed. Then the shaking force and shaking moment balancing are considered. In the last part a numerical simulation carried out by using ADAMS software illustrates that the balanced manipulators transmit no inertia loads to their bases. T parallel robots having between three and six DOFs, which was recently developed at the I.N.S.A. of Rennes (France) [START_REF] Briot | Analysis and optimization of a new family of parallel manipulators with decoupled motions[END_REF][START_REF] Arakelian | Design and prototyping of new 4-, 5-and 6-DOF parallel manipulators based on the copying properties of the pantograph linkage[END_REF][START_REF] V. Arakelian | Parallel robot comprising means for setting in motion a mobile element split in two separate subassemblies[END_REF][START_REF] Briot | PAMINSA: A new family of partially decoupled parallel manipulators[END_REF]. The particularity of this family of manipulators is the decoupling of the displacements of the mobile platform in the horizontal plane from the translations along the vertical axis. Fig. 1a shows a CAD model of a new PAMINSA architecture with 6 DOFs. The parallel structure of this manipulator has only three DOF for displacements of the platform along the x, y and z axes. With regard to the orientation of the end-effector, a serial wrist with 3 DOF is mounted on the platform. As in the previous architectures of PAMINSA manipulators, each leg of the PAMINSA is realized by a pantograph mechanism (Fig. 1b) with two input points A i and B i , and one output point C i (i = 1, 2, 3). Each input point A i is connected with the revolute actuator M i by means of a prismatic guide mounted on a rotating link. This type of architecture allows the generation of translations in the horizontal plane by use of the rotating actuators M 1 and M 3 and the vertical translations by the linear actuator M v . Thus, the displacements (x, y) of point P of the platform in the horizontal plane Oxy (respectively the translations along the x-and y-axes) are independent of the vertical translations z.

This implies that kinematic models controlling the displacement of the manipulator can be divided into two parts: -one model for the displacements in the horizontal plane (Fig. 2a); this model is equivalent to a 3-RPR manipulator with two actuated revolute joints and two parallel legs. On this drawing, the displacement of pairs M' i (or H' i ) corresponds to the displacement of both passive pairs M i (or H i ) and the pantograph linkage; -one model for the translations along the vertical axis (Fig. 2b) equivalent to the model for the vertical translations of a pantograph linkage (k is the magnification factor of the pantograph linkage).

The advantages of such a design concept have been discussed in [START_REF] V. Arakelian | Parallel robot comprising means for setting in motion a mobile element split in two separate subassemblies[END_REF][START_REF] Briot | PAMINSA: A new family of partially decoupled parallel manipulators[END_REF].

The PAMINSA manipulators are aimed at high-load carrying applications, but could also be used in different applications, such as pick-and-place or micro-manipulation (as long as the magnification factor of the pantograph linkages does not scale up the displacements but, on the contrary, scales down the movement).

As depicted above, the parallel structure has only three translational DOFs. In order to ensure the three orientational DOFs, a serial wrist is added to the platform (Fig. 3). This wrist has three rotations which can be represented by expressing the direction cosines in terms of z 0 -x 1 -z 2 Euler angles This wrist will be used in order to balance the shaking moments during the displacements of the structure. The positions of the centers of masses S ji of the links (j = 1 to 8 and i = 1, 2, 3) are considered at the middle of each link. Moreover, it is assumed that the centers of masses S wi of the wrist links are located at point P.

Therefore, for this mechanism, the shaking force F and shaking moment M applied on the base can be computed using the general equations:

     3 1 8 1 i j Sji ji P pl m m Γ Γ F , (1)
  rot 3 1 8 1 rot wrist i j Sji ji ji ji P pl m m M Γ OS M Γ OP M          (2)
with

ji ji ji ji ji Ω I Ω I M     rot (3) and     3 1 rot i wi wi wi wi wrist Ω I Ω I M   (4)
where -m pl represents the mass of the platform with the wrist; - P represents the acceleration of the end-effector; -m ji and I ji , represent the mass and the inertia matrix (expressed in the base frame) of the j-th limb of leg i, respectively; - Sji represents the acceleration of the centre of masses of the j-th limb of leg i; - ji is the vector of rotational velocities of the j-th limb of leg i; -I wi represents the inertia matrix expressed in the base frame of the i-th link of the wrist; - wi is the vector of rotational velocities of the i-th link of the wrist.

Developing and simplifying (1), the expression of the force F can be rewritten under the form:

                         3 1 5 5 2 1 1 i i S i S Ei Ei Ci Ci Ci Ci m m m m m Γ Γ Γ F       (5)
where  Ci ,  Ei ,  S5i represent the acceleration of points C i , S 5i and E i , and:

k m m m m m m m i i i i i pl Ci 2 2 2 2 2 3 7 6 4 2 1 2       ( 6
) ) 1 ( 2 2 2 / ) 1 ( 2 3 8 4 2 2 1 4 3 1 k k m m m k m m k m m m m i i i i i i i pl Ci           (7)   k m m m k m k m m m i i i i i i Ei 2 2 2 1 2 8 4 2 2 3 1        ( 8
) i i S m m 5 5  (9)
where k = B i F i /D i F i is the magnification factor of the pantograph linkage.

In order to estimate the variable dynamic loads transmitted to the base of PAMINSA, we carried out the simulations using ADAMS software. For this purpose, a standard cycle of pick-and-place motion (25×300 mm in 0.5 s) for a fifth-order polynomial trajectory with the following initial and final locations x i = 0.48 m, x f = 0.48 m and y i = -0.1 m, y f = 0.2 m with constant orientation of the wrist equal to zero. The geometric and inertia parameters of the robot are the following:

parameters for the position of points M i and J i : a = 0. terms of the inertia matrices of the links of the wrist:

B i G i = 0.442 m, A i E i = D i F i = 0.42 m, C i E i =k A i F i = 0.63 m, C i J i = 0.0275 m, M i S 5i = 0.15 m, M i O i = 0.
2 ) 1 ( kg/m 019 . 0  w XX J , 2 ) 1 ( kg/m 337 . 0  w YY J , 2 ) 2 ( kg/m 039 . 0  w XX J , 2 ) 2 ( kg/m 63 . 2  w YY J , 2 ) 3 ( kg/m 088 . 0  w XX J , 2 ) 3 ( kg/m 35 . 3  w YY J
. Fig. 4 shows the shaking force and moment variations of the unbalanced robot computed with ADAMS (in solid line).

B. Shaking force and shaking moment balancing

The shaking force can be balanced by cancelling the terms m Ci1 , m Ci2 , m S5i and m Ei . This can be achieved by the redistribution of the centres of masses of the moving links, i.e.: -for the cancellation of terms m Ci1 and m Ei , by the addition of counterweights m CP1i and m CP2i positioned, by example, at points A i and F i respectively, -for canceling the effects of term m S5i , by adding a counterweight m CP3i on the link 5 i positioned, by example, at points L i such as OL i = OO i + r O i S 5i , r < 0. -for canceling the effects of term m Ci2 , by adding counterweights undergoing similar but opposite movements to the end-effector motion.

With the added counterweights and the supplementary system, equations ( 6) to ( 9) can be rewritten as:

) 1 ( 2 2 2 2 / ) 1 ( 2 2 3 2 8 2 4 2 1 1 4 3 1 k k m m m m k m m k m m m m m i CP i i i i i i CP i i pl Ci             (10) 2 3 2 2 2 2 2 1 2 7 6 4 2 2 i pl i CP i i i i Ci m m k m m m m m m        (11)   k m m m m k m k m m m i CP i i i i i i Ei 2 2 2 2 1 2 2 8 4 2 2 3 1         (12) i CP i i S m r m m 5 5 5

 

(13) which leads to the expressions of the masses of the counterweights:

k m m m m m k m m k m m k m i CP i i i pl i i i i i CP 2 2 2 3 2 / ) 1 ( 2 ) 1 ( 2 8 2 4 2 1 4 3 1                    (14)
               k m m m k m k m m k m i i i i i i i CP 2 2 2 1 2 8 4 2 2 3 1 2 (15) r m m i i CP / 5 3   . ( 16
)
The effects of counterweights on shaking force and shaking moment are shown in Fig. 4 (with a dashed line). For the manipulator under study, the masses of the counterweights are m CP1i = 4.49 kg, m CP2i = 1.69 kg, and m CP3i = 0.436 kg (taking into account that r = -1). Note that, although the added masses allow cancelling of the shaking force, there is an increase of about 35% in the shaking moment around the z-axis. Concerning the shaking moment around x and y axes, the values before and after mass balancing are similar.

After force balancing, the expression of the shaking moment M bal becomes:

               3 1 3 3 1 2 1 i Li i i CP i Fi i i CP Ai i i CP bal m m m    OL OF OA M M (17)
The expression of M is given in equation [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF].

The term M of equation ( 17) depends on a free parameter rot wrist

M

, which defines the orientations of the end-effector between its initial and final positions, but which does not have any influence on the prescribed trajectory.

Thus, we have to adjust the rotations of the wrist in order to cancel the shaking moment. However, according to equation [START_REF] Wu | On the synthesis of a reactionless 6-DOF parallel mechanism using planar four-bar linkages[END_REF], which is nonlinear, the expressions of the rotations are difficult to compute. Therefore it is preferable to compute the values of the rotation angles starting from the

where -V P represents the velocity of the end-effector, -V Sji represents the velocity of the centre of masses of the j-th limb of leg i.

For reason of brevity, their expressions are not given in here.

In order to cancel the shaking moment of the mechanism, its angular moment H has to be constant. Thus, considering that H should be equal to zero, the following condition will be achieved: , one can rewrite this term as a linear combination of the angular velocities of the actuators of the wrist: [START_REF] Arakelian | Dynamic Balancing of the Scara Robot[END_REF] where I wi is the inertia matrix of the i-th link of the wrist, 0 ij corresponds to the zero matrix with i lines and j columns, E 3 is a rotation matrix:

  T wrist       N H  rot (22) with   T w w w w                                    0 1 0 1 0 0 0 0 0 0 0 1 1 2 1 2 1 2 1 2 1 1 2 2 2 0 0 0 I E I I I 0 0 0 N 3 3 3 2 1
) , ( ) , ( 3 x z   Rot Rot E  . ( 24
) Thus, by introducing equation [START_REF] Arakelian | Dynamic Balancing of the Scara Robot[END_REF] into relation [START_REF] Papadopoulos | Design and motion planning for a zero-reaction manipulator[END_REF], it is possible to determine the trajectory of the wrist, which can cancel the angular momentum, and as a result, the shaking moment:

D N 1 ] , , [    T       (25
) Taking into account that matrix N depends on the angles of rotation of the wrist, the value of  and  at time t can be found by a numerical integration: where t corresponds to the integration step, and the values of the angles and angular velocities at time t -t are considered known.

t t t t t t t t t t t t t t t t                                                       ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (    (26)
For the examined manipulator, we determine the required rotations of the end-effector, which makes it possible to cancel shaking moment (Fig. 5).

It should be noted that the moment of inertia of link 3 of the wrist is relatively small compared to the moments of inertia of the robot links. Thus, to ensure the balancing of the shaking moment, it is necessary that it makes several full revolutions during the motion of the robot in the horizontal plane.

Finally, note that in practice there are small deviations from the ideal zero reaction. Sources of such deviations include manufacturing errors, friction and clearances in the joints and drive actuators, as well as manipulator unbalancing due to a payload. However, in all of these cases, the proposed balancing method will eliminate the most significant base reactions due to the accelerating links of the robot.

III. CONCLUSION

A new field of shaking force and shaking moment balancing is in the development of fast parallel manipulators, which are very efficient for advanced robotic applications. In this paper, a shaking force and shaking moment balancing solution is developed for the positionorientation decoupled PAMINSA manipulator. The shaking forces of the manipulator in the horizontal plane are eliminated via optimum redistribution of masses, making the total mass center of the moving links stationary. With regard to the vertical inertia forces, the opposite motion of the endeffector and counterweights is applied. The shaking moment on the frame of the manipulator are eliminated by optimal control of the decoupled orientation motion. Numerical simulations carried out using ADAMS software show that after such a balancing, the manipulator transmits no inertia loads to its surroundings, i.e., the sum of all ground bearing forces and their moments are eliminated.

Fig. 1 .

 1 Fig. 1. The hybrid PAMINSA manipulator. (a) PAMINSA with 6 DOF. (b) Kinematic chain of each leg of the PAMINSA structure

  1 m, b = 0.175 m, c = 0.42 m, d = 0.1 m, e = 0.05 m and f = 0.12 m ; -magnification factor of the pantograph: k = 3; -lengths of the links of the pantograph linkages:

  308 m; -mass of the platform with the wrist: m pl = 4.41 kg; -masses of the links of the pantograph linkages: m 1i = 0.622 kg, m 2i = m 3i = 0.406 kg, m 4i = 0.107 kg, m 5i = 0.436 kg, m 6i = 0.921 kg, m 7i = 1.2 kg, m 8i = 0.305 kg; -terms of the inertia matrices of the links of the pantograph linkages:

Fig. 4

 4 Fig. 4 Shaking force before (solid line) and after (dashed line) balancing and shaking moment before (solid line) and after force-balancing (dashed line). (a) Shaking force along x-axis. (b) Shaking force along y-axis. (c) Shaking force along z-axis. (c) Shaking moment around x-axis. (e) Shaking moment around y-axis. (f) Shaking moment around z-axis.

Fig. 5 .

 5 Fig. 5. Rotations of the wrist used for canceling the shaking moment. (a) Angle . (b) Angle . (c) Angle ..

Complete Shaking Force and Shaking Moment Balancing of the Position-Orientation Decoupled PAMINSA Manipulator Sébastien Briot and Vigen Arakelian

	II. POSITION-ORIENTATION DECOUPLED PAMINSA
	MANIPULATOR WITH 6-DOFS
	A. Shaking force and shaking moment
	PAMINSA (PArallel Manipulator of the INSA) mani-
	pulators belong to a new family of partially decoupled


Sébastien Briot
email: sebastien.briot.1@ens.etsmtl.cailian.bonev@etsmtl.ca

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

SELF MOTIONS OF THE PANTOPTERON

In this paper, the self motions of a novel 3-DOF fully decoupled translational parallel robot, called the Pantopteron, are presented. The Pantopteron is similar to the well-known Cartesian parallel manipulator (Tripteron), but due to the use of pantograph linkages, an amplification effect is achieved. Therefore, equipped with the same actuators, the mobile platform of the Pantopteron moves faster than that of the Tripteron. This amplification is defined by the magnification factor of the pantograph linkages. The self motions are probably the most critical types of singularities a manipulator can meet. Therefore it is of utmost importance to have a good knowledge of them. Design considerations are also discussed in order to create Pantopterons without self motions.

INTRODUCTION

From an industrial point of view, the complexity and the existence of numerous singular configurations seem to be the worse drawback of parallel robots because these configurations reduce the workspace, which is already smaller than that of similarly-sized serial robots. Fortunately, the determination of singularities is a well studied problem and several methods have already been presented [START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF][START_REF] Bonev | Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory[END_REF][START_REF] Zlatanov | Singularity Analysis of Mechanisms and Robots via a Velocity-Equation Model of the Instantaneous Kinematics[END_REF].

The worst singular configuration a parallel robot can meet is the Type 2 singularity [START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF]. In such a singularity, the robot gains at least one degree of freedom and cannot resist some wrenches applied to its platform. Furthermore, the robot cannot exit such a singular configuration, without external help. Type 2 singular configurations can be divided into two classes, depending on the nature of the degree(s) of freedom gained, being either infinitesimal or finite, i.e., self-motion. However, merely studying the Jacobian matrix of the robot [START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF][START_REF] Bonev | Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory[END_REF], one cannot identify the nature of Type 2 singularities. Symmetry and, more precisely, design conditions that simplify the generally too complex direct kinematics of parallel robots are often privileged by robot designers. Unfortunately, such design conditions usually lead to self-motions, which are certainly the worst type of singularity. Hence, it is essential that the design conditions for such self-motions be well known in order to be avoided.

Several papers discuss self-motions in parallel robots. Not surprisingly, most of them deal with the Gough-Stewart platform whose direct kinematic model leads to as many as forty real solutions, for a relatively general design. Design conditions simplifying the direct kinematics of the Gough-Stewart platform, and subsequently leading to self-motions, are given in [START_REF] Karger | Singularities and Self-motions of Equiform Platforms[END_REF][START_REF] Karger | Architecture Singular Planar Parallel Robots[END_REF][START_REF] Husty | A Special Type of Singular Stewart-Gough Platform[END_REF][START_REF] Husty | Self-Motions of Griffis-Duffy Type Parallel Robots[END_REF][START_REF] Wohlhart | Mobile 6-SPS Parallel Robots[END_REF]. A classification of all self-motions of the Stewart-Gough platform is presented in [START_REF] Karger | Classification of all Self-Motions of the Original Stewart-Gough Platform[END_REF]. It is shown that the selfmotions can be translations, pure rotations, generalized screw motions, motions equivalent to the displacements of spherical four-bar mechanisms, or more complex spatial motions.

The Stewart-Gough platform is not the only parallel robot with self-motions. A few other parallel robots having selfmotions have also been studied. For example, in [START_REF] Bonev | Working and Assembly Modes of the Agile Eye[END_REF], it is shown that all singularities of the special 3-RRR (R stands for a passive revolute joint and R for an actuated revolute joint) spherical parallel robot, known as the Agile Eye, are selfmotions. The analysis of self mobility of spatial 5R closedloop mechanisms with one degree of freedom are presented in [START_REF] Karger | Classification of 5R Closed Kinematic Chain with Self Mobility[END_REF]. Reference [START_REF] Bandyopadhyay | Analysis of Configuration Space Singularities of Closed-Loop Mechanisms and Parallel Robots[END_REF] discusses the determination of generalized analytical expressions for the analysis of self-motions and presents several examples for both planar and spatial mechanisms with legs composed of R joints. The self motions of planar 3-RPR manipulators are also studied in [START_REF] Chablat | Self-Motions of a Special 3-RPR Planar Parallel Robot[END_REF][START_REF] Briot | Self-Motions of General 3-RPR Planar Parallel Robots[END_REF].

In this paper, we will study the self-motions of the so-called Pantopteron [START_REF] Briot | Pantopteron: a New Fully-Decoupled 3-DOF Translational Parallel Robot for Pickand-Place Applications[END_REF]. The Pantopteron is a manipulator similar to the Cartesian parallel manipulator (Tripteron) [START_REF] Gosselin | Cartesian Parallel Manipulators[END_REF], but due to the use of three pantograph linkages, an amplification of the actuators displacements is achieved. Therefore, given the same actuators, the mobile platform of the Pantopteron moves faster than that of the Tripteron. This amplification is equal to the magnification factor of the pantograph linkages.

The paper is organized as follows. Next, the architecture of the Pantopteron is briefly recalled. Then, a constraint analysis is presented, which will lead to the determination of the self motions. These singularities are kinematically described and various design considerations for the avoidance of self motions are given. Finally, conclusions are drawn.

DESCRIPTION OF THE ARCHITECTURE

The architecture of the Pantopteron is schematized in Fig. 1. It is composed of three identical legs which correspond to pantograph linkages (Fig. 2).

The pantograph is a mechanical system with two input points, A i and B i , and one output point C i (in the remainder of this paper, i = 1, 2, 3). These input points linearly control the displacement of the output point C i . A kinematic analysis shows that a linear actuator connected with input point B i controls the vertical displacement of the output point C i and another linear actuator with an axis parallel to a 1i controls the displacements along the same axis. Note that these motions are completely decoupled, i.e., they can be carried out independently. The input-output relationships for displacements are linear and are determined by the magnification factor k of the pantograph (k = A i C i /A i B i ). These properties of the pantograph mechanism are used in the Pantopteron For the Pantopteron, the actuators are located at the linear joints (1i) (Fig. 2). These three pairs are connected to the base so that their axes are orthogonal. All other joints are passive. Each pantograph linkage is attached to the platform at point C i via a Cardan joint, the axes each joint (12i) being orthogonal.

Figure 2. Schematics of a leg of the Pantopteron

They also are connected to actuator (1i) via a revolute joint, which allows the leg to have five DOFs, three translations and two rotations about the axes of the Cardan joint located at C i .. Such an architecture allows three fully-decoupled translational DOFs and its kinematics can be written under the form:

q J x    , (1) where 
          k k k 0 0 0 0 0 0 J . ( 2
)

Figure 1. Schematics of the Pantopteron

k is the magnification factor of the pantograph linkages, x  is the vector of the platform velocities and q  the vector of the actuator velocities (see [START_REF] Briot | Pantopteron: a New Fully-Decoupled 3-DOF Translational Parallel Robot for Pickand-Place Applications[END_REF] for further details).

CONSTRAINTS ANALYSIS

Constraints applied on the platform

Let x, y, z be the axes of the base frame (Fig. 1). It can be shown (see [START_REF] Briot | Pantopteron: a New Fully-Decoupled 3-DOF Translational Parallel Robot for Pickand-Place Applications[END_REF]) that each leg applies two wrenches on the platform:

-a moment r 1i (i = 1, 2, 3), which is orthogonal to the twist of all pairs (when actuators are disconnected); -a force r 2i which is due to the actuator M i . Let us consider leg 1. We denote by e j (j = 1 to 5) the unit screw corresponding to the passive displacement of the platform. The expression of these screws, expressed in the base frame at point C 1 , can be written as:

-for the translations along x, y and z,

  T 0 0 1 0 0 0 11  e ,   T 0 1 0 0 0 0 21  e and   T 1 0 0 0 0 0 31  e
; -for the rotations about the axes of joints (12,1) and [START_REF] Karger | Classification of 5R Closed Kinematic Chain with Self Mobility[END_REF][START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF],

  T 0 0 0 sin sin sin cos cos 1 1 1 1 1 41       e and   T 0 0 0 cos sin 0 1 1 51     e
, where  1 is the angle between a 11 and y axes, and  1 represents the rotation between vector a 31 and the axis of joint (12,1). The Plücker coordinates of the unit screws can be described in matrix

   y r (5b) 1 1 11 cos sin    z r (5c)
Thus, r 11 is a wrench of zero pitch (a pure moment). Similarly, it is possible to find that the wrenches r 1i transmitted to the platform by the legs when all actuators are disconnected are all pure moments. Let Q be the matrix composed of these wrenches applied on the platform by the legs. The expression of Q in the base frame, and expressed at point O, is: The passive displacements of the platform are orthogonal to this matrix of rank equal to 3. In the general case, there are three independent passive displacements, which are the three translations about x, y and z axes. Thus, the platform is constrained by the legs to have only translational displacements.

As joints (12,i)

  T 0 0 0 sin sin sin cos cos 1 1 1 1 1       41 e (10a)   T 0 0 0 sin cos cos sin sin 2 2 2 2 2       42 e (10b)   T 0 0 0 cos sin sin sin cos 3 3 3 3 3       43 e (10c)
Developing and simplifying Eqs. ( 9) leads to: 0 sin cos sin sin cos sin cos sin sin cos

2 2 1 1 2 1 1 2 2 1 1               f (11a) 0 sin cos sin sin cos sin cos sin sin cos 3 3 2 2 3 2 2 3 3 2 2               f (11b) 3 3 1 1 3 1 1 3 3 1 3 sin cos sin sin cos sin cos sin sin cos              f (11c)
Thus it is possible to find the expression of angles  i as a function of angles  i . Two distinct sets of solutions can be found:

0 3 2 1       (12) or              3 1 2 1 3 2 3 2 1 3 2 1 1 1 cos cos sin sin sin cos sin sin sin cos cos cos tan              (13a)              2 1 3 2 1 3 3 2 1 3 2 1 1 2 cos cos sin sin sin cos sin sin sin cos cos cos tan              (13b)              3 2 3 1 2 1 3 2 1 3 2 1 1 3 cos cos sin sin sin cos sin sin sin cos cos cos tan              (13c)
These two sets of solutions depend on the assembly of the mechanism. In a practical way, all angles  i are set to 0°.

However, it will be seen later that the self motions of the mechanisms can be found by the analysis of Eqs. [START_REF] Chablat | Self-Motions of a Special 3-RPR Planar Parallel Robot[END_REF].

Let us now consider that the actuator M 1 located at joint (1,1) is fixed. So, due to the decoupling properties of the pantograph linkages, the position about x axis of point C 1 is fixed. Thus, the platform has now two passive translational DOFs, which are orthogonal to the x axis. Therefore, a supplementary constraint is applied on the platform, which restrains its displacement.

Using an approach similar to the previous one, the second wrench applied by the leg on the platform, expressed at point where x Ci , y Ci and z Ci are the expressions of the coordinates of points C i along x, y and z axes.

C 1 , is   T 0 0 1 0 0 0 21  r . Let

Degeneracy of constraints

As the mechanism is fully isotropic, it does not have any Type 2 singularities. However, it has been shown in [START_REF] Briot | Pantopteron: a New Fully-Decoupled 3-DOF Translational Parallel Robot for Pickand-Place Applications[END_REF] that it has constraint singularities [START_REF] Zlatanov | Constraint Singularities of Parallel Mechanisms[END_REF]. These singular configurations appear if the system of wrenches applied on the platform degenerates, i.e., if the matrix R of Eq. ( 14) becomes singular. Such a singularity is called a constraint singularity. This happens if:

0 sin sin sin cos cos cos ) det( 3 2 1 3 2 1          R (15)
Introducing this expression in Eqs. [START_REF] Chablat | Self-Motions of a Special 3-RPR Planar Parallel Robot[END_REF], it can be seen that, in a constraint singularity, all angles  i may admit only one solution (indeed a double solution), which is  i = 0. This is true if the denominators of the Eqs. ( 13) are different from 0. If not, expressions (13) are indefinite, which implies that the system of Eqs. [START_REF] Karger | Classification of 5R Closed Kinematic Chain with Self Mobility[END_REF] degenerates. Thus, let us analyze the degeneracy of Eqs. [START_REF] Chablat | Self-Motions of a Special 3-RPR Planar Parallel Robot[END_REF]. These solutions degenerate if the following system of admits solutions: 0 sin sin sin cos cos cos

3 2 1 3 2 1 1          h (16a) 0 cos cos sin sin sin cos 3 1 2 1 3 2 2           h (16b) 0 cos cos sin sin sin cos 2 1 3 2 1 3 3           h (16c) 0 cos cos sin sin sin cos 3 2 3 1 2 1 4           h (16d)
It can be shown that this system of equations has several sets of solutions, which are:

  n   0 1 and 2 / 2     , n = 0, 1, 2… (17a) or 
 n   0 2 and 2 / 3     , n = 0, 1, 2… (17b) or 
 n   0 3 and 2 / 1     , n = 0, 1, 2… (17c)
Let us consider the case of Eq. (17a) (Fig. 3). Introducing these new conditions into the system of Eqs. ( 11) leads to the following set of solutions:

0 3   and    n    2 1 , n = 0, 1, 2… (18)
Thus, in such a case, the values of angles  1 and  2 are inde- terminate. Note that the same solutions for  2 and  3 or  1 and  3 could be found by using Eqs. (17b) and (17c), respectively.

In the next part, we will analyze the kinematics of the manipulator in such singular configurations.

SELF MOTION ANALYSIS

Kinematic analysis

Let us examine the manipulator when  1 = 0 and  2 = /2 (Fig. 3). In such a case, vectors a 2i of legs 1 and 2 of the manipulator are collinear, i.e., the planes of the pantograph linkages are parallel (here, they are horizontal). From Eq. ( 18),  3 = 0, therefore the altitude of the platform (and consequently of points C 1 and C 2 of the pantograph linkages, so as the values of angles  1 and  1 ) cannot change if actuator M 3 is fixed.

As the planes of the pantograph linkages stay horizontal, the passive prismatic joints of legs 1 and 2 are also horizontal plane (but also perpendicular), which implies that point C 1 can have a passive translation along the y axis, and point C 2 a passive translation along the x axis. Moreover, as the rotation  3 of joints (2,3) and (10,3) of leg 3, such as the displacement of the prismatic joint (9,3), are not constrained (these joints passive), the planar displacement of point C 3 of the platform is completely free.

Figure 4 represents the planar view of the manipulator in such a singularity. For clarity, leg 3 is replaced by an equivalent passive RP chain which completely allows the same planar displacement as the pantograph linkage.

As points C 1 and C 2 of the platform are constrained to move along lines L 1 and L 2 , respectively, and as the planar displacement of point C 3 is free, the platform undergoes a socalled Cardanic (self) motion [START_REF] Chablat | Self-Motions of a Special 3-RPR Planar Parallel Robot[END_REF][START_REF] Briot | Self-Motions of General 3-RPR Planar Parallel Robots[END_REF][START_REF] Sekulie | Method of Synthesis of Cardanic motion[END_REF][START_REF] Tischler | A Spatial Extension of Cardanic Movement: Its Geometry and some Derived Mechanisms[END_REF].

As a consequence, each point Q of the platform describes a curve E(Q), which may be:

-a straight line; -a doubly traced segment; -an ellipse.

The same type of self motions will be found by analyzing the conditions (17b) and (17c), but the movements will take place in other planes. In the next section, we will analyze the conditions on the design parameters that allow creating mechanisms without self motions.

Design considerations

Let us analyze once again the Eq. (17a), which can also be written under the form:

0 sin 1   and 0 cos 2   . ( 19
)
In the Cartesian space, Eq. ( 19) can be expressed as:

0 ) ( 1 1 1    G C z z and 0 ) ( 2 2 2    G C z z ( 20
)
where

2 1 1 2 1 1 1 ) ( ) ( G C G C z z y y      (21a) 2 2 2 2 2 2 2 ) ( ) ( G C G C x x z z      (21b)
and x Ci , y Ci and z Ci (x Gi , y Gi and z Gi , resp.) are the expressions of the coordinates of points C i (G i , resp.) along x, y and z axes. Disregarding the case where  i tends to infinity, singularities appear when:

0 1 1   G C z z and 0 2 2   G C z z ( 22
)
Taking into account that the terms x Gi , y Gi , z Gi appearing in Eq. ( 22) are constant and that

T i i i T i c b a z y x ] , , [ ] , , [   OC ( 23
)
where a i , b i , c i are constant terms, Eq. ( 22) can be rewritten under the form:

0 2 2 1 1       G G z c z z c z . ( 24
)
Thus, condition (17a) can appear if:

2 2 1 1 G G z c z c    . ( 25
)
By a similar way, it can be proven that condition (17b) may appear if:

2 2 1 1 G G x a x a    , ( 26
)
and condition (17c) if:

2 2 1 1 G G y b y b    . ( 27
)
Thus, by avoiding these three conditions, the mechanism will never have self motions.

CONCLUSIONS

The singularities of a novel 3-DOF fully-decoupled isotropic translational parallel mechanism, called the Pantopteron, were analyzed in detail. It was shown that the only sin-gularities that this robot can undergo are so-called constraint singularities. Furthermore, all of these constraint singularities were shown to be Cardanic self-motions. Fortunately, simple design conditions for eliminating the possibility for such singularities were found.

11 ,

 11 transmitted to the platform by the leg, is orthogonal to the twists composing the lines of matrix E 1 :

  are all orthogonal to each other, the following relations must hold:

  us denote by R the matrix composed of these wrenches applied on the platform by the legs. The expression of R in the base frame, and expressed at point O, is:

Figure 3 .

 3 Figure 3. Example of a constraint singularity for  1 = 0 and  2 = 90°.

Figure 4 .

 4 Figure 4. Cardanic self motion of the Pantopteron


V A Glazunov

Nguyen Minh Than
email: minhthanhnguyen@vnn.vn

paper xxCK2005 ON NEW PARALLEL-CROSS MECHANISMS

Keywords: Parallel mechanism, parallel-cross mechanism, workspace, pressure angle, singularity

This paper addresses the new parallel-cross mechanisms, their classification and analysis. Several crossing kinematic chains of these mechanisms are located between the parallel kinematic chains. The contribution concerning to these mechanisms is their more large workspace than this of parallel mechanisms.

Introduction

It is well known that the workspace of the parallel manipulators can be less than this of the serial manipulators because of interference of the links of the kinematic chains or because of singularities (see for example the works of Gough [START_REF] Gough | Contribution to Discussion to Papers on Research in Automobile Stability and Control and in Tyre Performance by Cornell Staff[END_REF], Stewart [START_REF] Stewart | Platform with Six Degrees of Freedom[END_REF], Hunt [START_REF] Hunt | Structural Kinematics of In-Parallel-Actuated Robot Arms[END_REF], Merlet [START_REF] Merlet | Parallel Robots[END_REF], Gosselin and Angeles [START_REF] Gosselin | Singularity Analysis of Closed Loop Kinematic Chains[END_REF], Parenti-Castelli and Innocenti [START_REF] Parenti-Castelli | Direct Displacement Analysis for Some Classes of Spatial Parallel Mechanisms[END_REF]). Besides, if parallel mechanism contains numerous kinematic chains then the construction of the base or of the moving platform can be rather complete due to necessity to situate on these links many kinematic pairs connecting the kinematic chains with the base and with the platform. One of the solutions of this problem is to situate two or more actuators in several kinematic chains so that the number of these chains will be less than six. However, in this way the architecture of the kinematis chains containing several actuators can be complete. In this paper, we develop other solution, which carried out by using mechanisms of parallel-cross structure (parallel-cross mechanisms), i.e. mechanisms in that some crossing kinematic chains are located between parallel kinematic chains (Glazunov et al. [START_REF] Glazunov | Development of manipulators with a parallel-cross structure[END_REF]). We represent the graphical classification of these mechanisms and their analysis comparing them with parallel mechanisms taking in account singularities and pressure angles, well known in the mechanism design (Sutherland and Roth [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF], Lin and Chang [START_REF] Lin | The force transmissivity index of planar linkage mechanisms[END_REF]) but not so often applied to the parallel mechanisms (see Arakelian et al. [10]).

Synthesis of parallel-cross structures

In this part, we would like to synthesize all possible structures with a number of degrees of freedom from 1 to 6, with at least one leg attached to two others (two legs must not be linked together by two others). In the following synthesis, each leg can be composed with at most 6 joints, and we consider that a leg with 6 joints has obligatorily one pair actuated.

The results are summed up in table 1. In this table, the notation ijkl-mn (with i, j, k, l, m, n = 2 to 6) corresponds to:

left term: the legs attached to both the fixed base and the moving platform with respectively i, j, k and l joints (passive of active); -right term: the legs attached to other legs with respectively m and n joints (passive of active).

3-3

In the following part, we propose to analyze two structures which can be synthesized by our approach and which have interesting characteristics.

Description of the proposed structure

Let us now consider the architecture of the new proposed mechanisms with 6 DOF (Fig. 1). Two cases can be considered in function of the position of the actuators: the actuators are located upon the base (Fig. 1a); the actuators are located under the base (Fig. 1b). Both manipulators are composed of one fixed base A 1 A 2 A 3 and one moving platform B 1 B 2 B 3 . The platform is linked to the base by the use of three legs, each of which composed of one spherical at point A i , one actuated prismatic pairs A i B i and one spherical joint B i . Moreover these legs are linked together by the use of three other linkages composed of one spherical joint C i , one actuated prismatic pair C i D j and one spherical joint D j (j = i + 1 for i = 1, 2, j = 1 for i = 3). It is obvious that such a mechanism contains intern mobility which can be constrained by the use of cardan joints (Fig. 1c). Moreover, as shown on Fig. 1c, the spherical joints can be replaced by three revolute joints. Considering a manipulator composed of 6 pris-matic, 6 spherical and 6 cardan joints, the Grübler formula can give the number of degrees of freedom m of the mechanism: Let us consider inverse geometric analysis. The positions of points A i , B i , C i , D i of the manipulators can be expressed as (i = 1, 2, 3):

  T b i i R 0 0 ) , ( 0 Rot OA z   (2)                       0 sin cos ) , ( ) , ( ) , ( i pl i pl i R R z y x      2 1 0 Rot Rot Rot OB y x z ( 3
) i i i i i l B A OA OC  1   ( 4
) i i i i i l B A OA OD  2   (5) with R b = OA i , R pl = PB i , ] 2 / , 6 / , 6 / 5 [        i , (i = 1, 2, 3) (as said previously, triangles A 1 A 2 A 3 and
B 1 B 2 B 3 are considered equilateral), l 1 = i i C A , l 2 = i i D
A , and Rot(, w) the matrix representing the rotation of angle  ( = ,  and ) around the w-axis of the intermediate frame (w = z 0 , x 1 and y 2 ). Thus the expressions of the articular coordinates are given by:

    i i T i i i B A B A   , for i = 1, 2, 3, (6)
    j i T j i i D C D C  3  , (j = i + 1 for i = 1, 2, j = 1 for i = 3). ( 7
)
Let us consider kinematic analysis. The closure equations are represented by the expressions:

    i i T i i i i f B A B A    2 0  for i = 1, 2, 3, (8)
    j i T j i i i f D C D C      2 3 3 0  , (j = i + 1 for i = 1, 2, j = 1 for i = 3). ( 9
)
Differentiating the closure equations with respect to time, one can obtain the following expression (Gosselin and Angeles [START_REF] Gosselin | Singularity Analysis of Closed Loop Kinematic Chains[END_REF]):

0 q B v A    (10)
where:

A and B are two matrices;

         i j X f A ,          i j f  B , X i = (x, y, z, , , ).
q  is the vector of the time derivatives of the coordinates of the actuators; T ] , , , , , [

6 5 4 3 2 1               q .
v is the vector of the time derivatives of the position of the platform;

T z y x ] , , , , , [           v .
The vector v can be related to the twist t of the platform expressed in the base frame using a transformation matrix D (Merlet [4]):

I

33 and 0 33 corresponds to identity and zero matrices of dimensions 33 respectively. Therefore the Jacobian matrix J relating the twist t of the platform to the velocities of the actuators is given by the relation:

B A D J 1    (12)
The elements of the matrices A and B depend on the parameters of the mechanisms and on the position of the moving platform. The next section deals with the comparison between the performances of the proposed manipulators and two other manipulators: the Stewart and Gough platforms.

Comparative analysis of performances

In order to appreciate the performances of the new proposed mechanisms, let us compare them with two other manipulators with 6 DOF which are well known, commonly used by manufacturers and also actuated by six prismatic pairs: the Gough platform (Fig. 3a) and the Stewart platform (Fig. 3b). It is obvious that the mechanisms presented on Fig. 3 contain intern mobility which can be constrained by the use of cardan or revolute joints (Fig. 3c). Their geometric and kinematic models are well known (see for example the works of Gough [START_REF] Gough | Contribution to Discussion to Papers on Research in Automobile Stability and Control and in Tyre Performance by Cornell Staff[END_REF], Stewart [START_REF] Stewart | Platform with Six Degrees of Freedom[END_REF], Hunt [START_REF] Hunt | Structural Kinematics of In-Parallel-Actuated Robot Arms[END_REF], Merlet [START_REF] Merlet | Parallel Robots[END_REF], Gosselin and Angeles [START_REF] Gosselin | Singularity Analysis of Closed Loop Kinematic Chains[END_REF], Parenti-Castelli and Innocenti [START_REF] Parenti-Castelli | Direct Displacement Analysis for Some Classes of Spatial Parallel Mechanisms[END_REF]) and, as computing the geometrics and kinematics of these two manipulators is not the main topic of this paper, we will not present them another time. So, we can consider the following problematic for our comparative analysis: starting from identical components (actuators, base and platform size), what are the arrangements of the actuators which will lead to the greatest workspace taking into consideration: the stroke of the actuators, the displacement of the spherical pairs and the interferences between the elements; the quality of effort transmission in the mechanism.

One industrial version of the Gough platform is the robot FANUC F200. This manipulator is well known for its high payload capacities and its ability to have little deformations under the application of a vertical force. Its approximate geometric parameters are:

-R pl = 0.2 m and R b = 0.3 m; -maximal stroke of the actuators (A i B i ) max -(A i B i ) min = 0.65 m. So in order to make a fair comparison, we propose to take into account the following design constraints:

-for each kind of manipulator, base and platform have to be identical to those of FANUC F200 robot; -the stroke of the actuators has to be identical and equal to 0.65 m; -in order to minimize the size of the manipulators, all points located on the base must be comprised on the surface defined by triangle A 1 A 2 A 3 . So now, let us analyse the size of the workspace taking into account the previous constraints. the considered manipulator, it should have at least one of the p configurations of the orientation space that valid the geometric limitations. Therefore, this workspace is defined as the maximal workspace (Merlet [4]).

The geometric limitations under study are, for each manipulator:

-the maximal stroke of the actuators has to be identical and equal to 0.65 m; -in order to avoid interferences between the different elements of the manipulators, the minimal distance between to non secant links has to be equal to 0.1 m, and we consider that the minimal angle between two secant links is equal to 15°. For example, on figure 3c, the angle between links A i D i and C i D i cannot be inferior to 15°; -the interval of rotation for the spherical joints has to be comprised between -60 deg and +60 deg. The geometric parameters for the Gough platform are already fixed. For the other manipulators, we chose the geometric parameters that are as close as possible to the parameters of the Gough platform:

-Stewart platform: points C 1 , C 2 and C 3 coincide with A 2 , A 3 and A 1 respectively, and A i D i = 0.2 m; -new proposed manipulator, case one (actuators upon the base): l 1 = 0.15, l 2 = 0.65; -new proposed manipulator, case two (actuators under the base): l 1 = -0.3, l 2 = -0.75. Fig. 4 presents the obtained workspaces. The sizes of the workspace are (m 3 ): Gough platform: 0.719; Stewart platform: 1.749; new proposed manipulator, case one: 1.959; new proposed manipulator, case two: 2.168. The next step of our analysis is the comparison the quality of effort transmission determined by pressure angles existing in the studied mechanisms. These angles can be defined as angles between vectors of forces and translational velocities of points at which the forces are applied. Note, that in Type 2 singular configurations (Gosselin and Angeles [START_REF] Gosselin | Singularity Analysis of Closed Loop Kinematic Chains[END_REF]) at least one of the pressure angles is equal to 90°. For any parallel mechanism with passive joints, it is well known that the i-th line of the inverse Jacobian matrix J -1 relating the twist of the platform expressed at point P to the particular velocities (

t J q 1   
) corresponds to the wrench w i applied by the i-th actuator on the platform. For a 6 DOF manipulator, the inverse Jacobian matrix J -1 can be written as:

J -1 = (w 1 ,. w 2 , w 3 , w 4 , w 5 , w 6 ) T (13) Without loss of generality, let us compute the expression of the twist t 1 of the platform when actuator 1 is dis- connected and the other actuators are fixed. t 1 is the twist reciprocal to the wrenches w 2 to w 6 applied on the platform by the actuators 2 to 6. Therefore we can have the five following relations:

iz z iy y ix x iz z iy y ix x M M M F v F v F v 1 1 1 1 1 1 0          , (i = 2 to 6) (14)
where: V P = [v 1x , v 1y , v 1z ] T represent the translational velocities of point P of the platform along the axes x, y and z respectively;

1 Ω  1x ,  1y ,  1z ] T represents the rotational velocities of the platform around the axes x, y and z respectively; For each of the four manipulators, the wrench applied by actuator 1 on the point B 1 of the platform is a pure force F 1 . The translational velocity V B1 of point B 1 can be deduced from the expression

1 1 P B1 Ω P B V V    .
The terms of this expression are detailed above. Therefore, the pressure angle  1 can be determined as:

          B1 1 B1 1 V F V F T 1 1 cos  (16)

Figure. 1 .

 1 Figure. 1. Possible architectures of the new proposed mechanism with three actuators (a) upon the base and (b) under the base, and (c) one possible design for the legs C i D j . Thus, the simultaneous displacements of actuated joints allow the mechanism having 3 translatory and 3 rotary DOF. We denote: the lengths of the prismatic pairs A i B i as  i , (i = 1 to 3); the lengths of the prismatic pairs C i D j as  i+3 , (j = i + 1 for i = 1, 2, j = 1 for i = 3). Triangles A 1 A 2 A 3 and B 1 B 2 B 3 are considered equilateral. We consider that the controlled point is the point P located at the centre of the triangle B 1 B 2 B 3 . The position of P is expressed by the vector [x, y, z] T . The base frame is centred at point O located at the centre of triangle A 1 A 2 A 3 (point O is not represented on Fig. 1). The x 0 axis of the base frame is directed along the line A 1 A 2 and the y 0 axis along the line OA 3 .The rotations of the platform are denoted , , which can be obtained by expressing the directional cosines in terms of z 0 -x 1 -y 2 Euler angles , (Fig.2).

Figure. 2 .

 2 Figure. 2. Description of the chosen Euler angles.

Figure. 3 .

 3 Figure. 3. Schematic representation of (a) the Gough platform and (b) the Stewart platform, and (c) one other possible design for the legs of the Stewart platform. Let us consider workspace analysis taking into account geometric limitations. The corresponding procedure is very simple. First of all, we discretize the Cartesian space into n points and the orientation space into p configurations. The step for the Cartesian space is equal to 0.05 m and for the orientation space to 10 deg for angles comprised between -60 deg and +60 deg for each angle ,  and . For a point Q to belong to the workspace of

2 Figure. 4 .

 24 Figure. 4. Workspaces of the manipulators under study.

F

  ix , F iy , F iz represent the forces applied on the platform by the i-th actuator along the axes x, y and z respectively; M ix , M iy , M iz represent the moments applied at point P on the platform by the i-th actuator around the axes x, y and z respectively. Fixing the value of v 1x to 1, one can determinate the expressions of the other terms of the twist t 1 :

Table 1 .

 1 Representation of all possible structures with at least one leg attached to two others.

			Number of degrees of freedom
	6	5	4	3	2	1

Table 2 .

 2 Schematic representation of results of table 1 for manipulators with legs as close as possible from the base.

			Number of degrees of freedom
	6	5	4	3	2	1

Please note that the pressure angle is an indicator of the power transmitted by the actuator 1 to the platform, which can be written as:

These pressure angles can be calculated in a similar way for the Gough and Stewart platforms and for actuators 1 to 3 of the new mechanisms. However, for actuators 4 to 6 of the new manipulators, the way to compute the pressure angles is a bit different. Without loss of generality, let us consider the link C 1 D 2 (Fig. 1a). Basing on relation, equivalent to (15), it is possible to compute the twist t 4 of the platform when actuator 4 is disconnected and the others are fixed, and as a result, the translational velocities V C1 and V D2 :

The power transmitted by the actuator to the mechanism can be written under the form:

, where F 4 represents the effort applied by the actuators on links A 1 B 1 and A 2 B 2 , V D2C1 represents the relative velocity of the point D 2 from the point C 1 . Therefore, one can define a pressure angle  4 :

The same approach can be used in order to determine the pressure angles of actuators 5 and 6. Thus, we can now analyse the workspace of the studied manipulators taking into account the maximal value of the pressure angles of the mechanisms. We consider that the maximal admissible value for the pressure angle is 82°. For one point of the workspace, if there exists at least one configuration of the orientation space for which the pressure angle is inferior to 82°, so this point belongs to the workspace of the robot. Therefore, we can obtain new results about the workspace of our manipulators. The sizes of the workspace are (m 3 ): Gough platform: 0.719; Stewart platform: 1.727; new proposed manipulator, case one: 1.952; new proposed manipulator, case two: 2.154.The new results are similar to those obtained previously.

Conclusion.

Thus, in this article, the parallel-cross manipulators are represented as well as their classification. The comparison analysis executed for the new manipulators and well known parallel manipulators taking into account geometric limitations and admissible pressure angles allows to conclude, that the workspace of the new manipulators is not less than this of well known parallel manipulators.