Arakelian

C Baradat

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

S Briot

S Guegan

Design and Prototyping of a New Balancing Mechanism for Spatial Parallel Manipulators

Keywords: balancing, torque compensation, parallel mechanisms, Delta robot

This paper proposes a new solution to the problem of torque minimization of spatial parallel manipulators. The suggested approach involves connecting a secondary mechanical system to the initial structure, which generates a vertical force applied to the manipulator platform. Two versions of the added force are considered: constant and variable. The conditions for optimization are formulated by the minimization of the root-mean-square values of the input torques. The positioning errors of the unbalanced and balanced parallel manipulators are provided. It is shown that the elastic deformations of the manipulator structure which are due to the payload, change the altitude and the inclination of the platform. A significant reduction of these errors is achieved by using the balancing mechanism. The efficiency of the suggested solution is illustrated by numerical simulations and experimental verifications. The prototype of the suggested balancing mechanism for the Delta robot is also presented.

Introduction

A mechanism of parallel architecture is statically balanced if its potential energy is constant for all possible configurations 1 . This means that the mechanism is statically stable for any configuration, i.e. zero actuator torques due to the static loads are required. For static balancing of robot mechanisms, different approaches and solutions have been developed and documented . The balancing schemes for robotic systems can be systematized by means of balancing (table 1): counterweight (group A), spring (group B), pneumatic or hydraulic cylinder, electromagnetic device, etc. (group C). Each group can be presented by the following subgroups: A1. Balancing by counterweights mounted on the links of the initial system [START_REF] Agrawal | Reactionless space and ground robots: novel designs and concept studies[END_REF][START_REF] Wang | Static balancing of spatial three-degree-of-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]. Such balancing is very simple to realize. However, it leads to the important increase of the moving masses of the manipulator, and as a result, its inertia. A2. Balancing by counterweights mounted on the auxiliary linkage connected with the initial system [START_REF] Fujikoshi | Balancing apparatus for jointed robot[END_REF][START_REF] Arakelian | Manipulator[END_REF][START_REF] Wang | Static balancing of spatial four-degree-of-freedom parallel mechanisms[END_REF][START_REF] Russo | Static balancing of parallel robots[END_REF]. Articulated dyads or pantograph mechanism are used as an auxiliary linkage. B1. Balancing by springs jointed directly with manipulator links [START_REF] Lakota | Manipulators for assembly tasks[END_REF][START_REF] Ebert-Uphoff | Static balancing of spatial parallel mechanisms -Revisited[END_REF][START_REF] Herder | Energy-free systems. Theory, conception and design of statically balanced mechanisms[END_REF][START_REF] Streit | Equilibrators for planar linkages[END_REF] B2. Balancing by using a cable and pulley arrangement [START_REF] Vrijlandt | Seating unit for supporting a body or part of a body[END_REF][START_REF] Vladov | Module of linear motion of industrial robot[END_REF][START_REF] Ebert-Uphoff | Practical considerations for the static balancing of mechanisms of parallel architecture[END_REF][START_REF] Tuda | Arm with gravity-balancing function[END_REF]. Such an approach allows zero free length springs to be used, which is more favorable for realization of complete balancing.

Balancing by using an auxiliary mechanism can be presented in the following manner: B3.1. Balancing by using an auxiliary linkage [START_REF] Herder | Some considerations regarding statically balanced parallel mechanisms[END_REF][START_REF] Leblond | Static balancing of spatial and planar parallel manipulators with prismatic actuators[END_REF][START_REF] Popov | Balanced manipulator[END_REF][START_REF] Agrawal | Design of gravity balancing leg orthosis using non-zero free length springs[END_REF][START_REF] Simionescu | The static balancing of the industrial arms. Part I: Discrete balancing[END_REF][START_REF] Segla | Statical balancing of a robot mechanism with the aid of a genetic algorithm[END_REF][START_REF] Minotti | Ressort et mécanismes : une solution aux problèmes d'équilibrage[END_REF][START_REF] Dzhavakhyan | Balanced manipulator[END_REF][START_REF] Hervé | Device for counter-balancing the forces due to gravity in a robot arm[END_REF][START_REF] Bartlett | Robot with spring pivot balancing mechanism[END_REF][START_REF] Popov | Counterbalanced manipulator[END_REF]. B3.2. Balancing by using a cam mechanism [START_REF] Simionescu | The static balancing of the industrial arms. Part I: Continuos balancing[END_REF][START_REF] Lakota | Manipulators for assembly tasks[END_REF][START_REF] Kondrin | Pivoted arm balancing mechanism[END_REF][START_REF] Petrov | Vertical displacement device[END_REF]. B3.3. Balancing by using gear train [START_REF] Popov | Balanced manipulator[END_REF][START_REF] Gvozdev | Manipulator[END_REF][START_REF] Gvozdev | Manipulator[END_REF][START_REF] Gvozdev | Manipulator[END_REF]. 1 It should be noted that, in the balancing of high-speed mechanisms, the term «static balancing» refers to shaking force cancellation or minimization [1,[START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF]. With regard to the «static balancing» in robotics, this term Corresponding author: Arakelian C. Balancing by using pneumatic or hydraulic cylinders, which are connected with manipulator links [START_REF] Belyanin | Balanced manipulators[END_REF] or directly with the moving platform [START_REF] Wildenberg | Compensating System for a Hexapod[END_REF]. There is a balancing approach based on counterweights, which are fluid reservoirs. Continuous balancing is achieved by the pumping of the fluid from the first reservoir-counterweight to the second [START_REF] Dzhavakhyan | Balanced manipulator[END_REF]. Electromagnetic effects were also used for balancing [START_REF] Segawa | Parallel Link Mechanism[END_REF].

The literature review showed that many balancing methods are applicable for planar parallel manipulators. However the balancing of spatial parallel architectures is a complicated problem because it can be achieved either by unavoidable increase of the total mass of moving links or by a considerably complicated design of the initial parallel mechanism. Let us consider this problem for the Delta robot.

The Delta robot [START_REF] Clavel | Device for movement and displacing of an element in space[END_REF] was developed for high-speed manipulation and it is well known in the electronics, food and pharmaceutical sectors as a reliable system with fast execution of light-duty tasks. However, in recent years, more attention has been paid to the increasing number of possible industrial applications, such as manipulation of medical devices (Fig. 1). In this case, the displacement speed of the platform is not essential because there is no need for productivity acceleration. However, as a result of the increased mass of the platform (about 70kg), the input torques became important. Thus it became evident that the platform's mass must be balanced. In this case, the traditional approaches with counterweights and springs mounted on the moving links are not applicable. The Delta robot has a complex structure and after such balancing it becomes either very heavy or a complicated assembly with several complementary articulated dyads. That is why another means for the solution of this problem is proposed. It consists in the addition of a secondary mechanism between the manipulator base and moving platform. This mechanism can create a supplementary vertical force F on the platform to balance the gravitational forces of the robot (Fig. 2).

In this context, a new balancing mechanism for the minimization of the input torque of the spatial parallel manipulators with high weight-carrying capacity is developed.

Description of the balancing mechanism

The suggested balancing system includes (Fig. 3) a pantograph mechanism mounted on the rotating stand connected with the base. Thus the position of point C is represented by vector P = [x, y, z] T and the passive motions of the pantograph are represented by q = [r, , Z] T . The kinematic relations between P and q are the following: x = (1-k) r cos, y = (1-k) r sin and z = k Z. Differentiating these equations with respect to time, one obtains:

P J q   1   (1)
where

  T z y x      P (2)   T Z θ r      q (3) Corresponding author: Arakelian                k θ r k θ k θ r k θ k 0 0 0 cos ) 1 ( sin ) 1 ( 0 sin ) 1 ( cos ) 1 ( J (4)
It is obvious that the added balancing system cannot follow all trajectories of the parallel robot. For example, if the given trajectory of the parallel robot is composed from two mutually perpendicular straight lines, which are intersected at the point of 0   y x , the balancing mechanism cannot execute a continuous motion. In this case, it is necessary to orientate the plane of the pantograph mechanism relative to the Z axis. Thus, it is evident that the balancing mechanism must be equipped with a complementary rotating actuator for its orientation in the case of singular trajectories. This complementary actuator may be in operation only for special cases. However, it is necessary to note that the suggested balancing mechanism is applicable to many spatial parallel robots from 3 to 6 degrees of freedom.

Fig. 5 shows an example of such an application for the Gough-Stewart platform with the implemented balancing system. In the proposed design of the Gough-Stewart platform the payload is balanced by the suggested mechanism. So the platform becomes a weightless link and it can be displaced and oriented by low-power linear actuators.

In the following section we consider the balancing of the Delta robot by means of the proposed mechanism and we discuss the minimization of the input torques by a constant or a variable force.

Minimization of the torque by a constant force applied to the robot platform

Let us examine two cases: minimization of the torques due to the static loads, i.e. weights of the moving links, and dynamic forces, i.e. inertia forces.

Minimization of the torques due to static loads (gravitational forces).

The input torque of the n th actuator can be expressed as:

st n st n st n st n st n M M M M M 4 3 2 1     (5)
where st n M 1 is the torque due to the gravitational forces of the arms (see Fig. 6),

st n M 2 is the torque due to the parallelograms, st n M 3 is the torque due to the joints on points B i , st n M 4 is the torque due to the gravity forces of the platform and medical device. For i, n =1, 2, 3 and for j =1, 2, 3 st jn Μ is equal to:

p i ji T ji st jn z y x z y x           3 1 ) , , ( ) , , ( G J
M [START_REF] Laliberté | Static balancing of 3-DOF planar parallel mechanisms[END_REF] and for n = 1, 2, 3 and j = 4, st n 4

Μ is equal to:   p T st p z y x z y x G J M   ) , , ( ) , , ( 4 (7)
where J 1i is the Jacobian matrix between the point P 1i and the actuated variables  n , J 2i is the Jacobian matrix between the point P 2i and the actuated variables n , J 3i is the Jacobian matrix between the point B i and the actuated variables n , J is the general Jacobian matrix of the robot, between the point P and the actuated variables  n for i, n = 1, 2, 3, and G and G ji are the gravity forces (Fig. 6). The matrix J ji (j = 1, 2, 3) can be written as: . ;

; ; The three input torques can be presented by the following expression:

3 2 2 1 1                                     n n i i n i i n i i     OP J OB J OP J OP J (8)
                                F M M M M M M T st st st st bal st bal st bal 0 0 3 2 1 3 2 1 J (9)
where st ibal M is the optimized torque of actuator i (i = 1, 2, 3).

The condition for the minimization of the root-mean-square (RMS) value of the torques can be expressed as:

  F N p i p p p i p p p st i N F z y x J z y x M min ) , , ( ) , , ( 1 3 1 2
) 3 (             (10)
where

st i
M is the initial torque of the actuator i, N is the number of calculated positions of the robot, J (3i) is the i th column of the third line of the matrix J, i = 1, 2, 3 is the number of the actuator and x p , y p , z p are the coordinates of the p th calculated position of the workspace.

For the minimization of the RMS value of the torques, it is necessary to minimize the sum: For this purpose, we shall achieve the condition:

               N k i p p p i p p p st i F z y x J z y x M 1 3 1 2 ) 3 ( ) , , ( ) , , ( (11)
0     F
, from which we determine the force:

                      N k i p p p i N k i p p p i p p p i z y x J z y x M z y x J F 1 3 1 2 ) 3 ( 1 3 1 ) 3 ( ) , , ( ) , , ( ) , , ( (12)
Numerical example. For the Delta robot of the SurgiScope ® , the parameters are the following (see Fig. 6):

l AiBi =0,75 m; l BiCi = 0,95 m; m 1i = 2,3 kg (mass of i th arm with center P 1i ); m 2i = 5,2 kg (mass of i th parallelogram with center P 2i ); m 3i = 3,1 kg (mass of the joint at point B i ); m = 79 kg (mass of the platform, joints and medical device, with center in the point P);

-l AiP1i = l AiBi /2 = 0,375 m; -l BiP2i = l BiCi /2 = 0,475 m.
Thus the value of the added force for the given parameters is F = 931 N. Fig. 7 shows the variations of the input torques for unbalanced and balanced Delta robot.

Thus, the reduction of the RMS value of the input torque is 99.5 %. The reduction of the maximum value of the torque is 92%.

The purpose of this study is to develop a reliable mechanism for gravitational force balancing of spatial parallel manipulators. Moreover, it is also tempting to consider the minimization of the torques due to the dynamic loads, i.e. inertia forces. Corresponding author: Arakelian

Minimization of the torques due to the dynamic loads (inertia and gravitational forces).

The input torque of the i th actuator can be expressed as [START_REF] Miller | The Lagrange-based model of Delta-4 robot dynamics[END_REF]:

3 , 2 , 1 , 3 ,                  i A L L dt d M i i i i i dyn i     . ( 13
)
With the added force F, Eq. ( 13) can be written as:

F A B M M i i i dyn i dyn ibal 3 , 3    (14)
where [B ij ] is the inverse matrix of matrix A [START_REF] Miller | The Lagrange-based model of Delta-4 robot dynamics[END_REF] composed by the first three columns only.

We would like to point out that, in the case of the dynamic study, the input torques depend on the velocity and acceleration of the platform displacement and it is impossible to realize an optimization for the whole workspace of the robot (as it was for the static load minimization).

Thus we must define a trajectory in which the input torques will be minimized. The selected trajectory is presented in Fig. 8. The kinematic characteristics of the examined motion are given by the maximum values of the acceleration and velocity and presented in table 2 (factors «a» and «v»).

The condition for the minimization of the RMS value of the torques can be expressed as: Corresponding author: Arakelian

  F N p i i i i dyn i N F A B M min 1 3 1 2 3 , 3              (15)
where

dyn i M
is the initial torque of the actuator i, N is the number of calculated positions of the robot, i = 1, 2, 3 is the number of the actuator.

For the minimization of the RMS value of the torques, it is necessary to minimize the sum:

  F N p i i i i dyn i F A B M min 1 3 1 2 3 , 3                (16)
For this purpose, we shall achieve the condition: 0     F , from which we determine the force:

                          N p i i i i N p i dyn i i i i A B M A B F 1 3 1 2 3 , 3 1 3 1 3 , 3 (17)
For the examined trajectory with a = 1,05 m/s² and v = 0,79 m/s, we determine the external force: F = 934 N.

Minimization of the input torques by a variable force applied to the platform of the robot

This section also contains two cases: minimization of the input torques due to the static and dynamic forces. Corresponding author: Arakelian

Minimization of the torques due to the static loads (gravitational forces).

The relationship between the actuator input torques and the resultant force can be written as:

st T res M J F   (18)
where

  T st st st st M M M 3 2 1  M .
This variable force has three components along the X, Y and Z axes. For minimization of the input torques we use the component of F res along the Z axis, which is similar to the added force F.

It should be noted that the difference in the minimized torques between the two examined cases (with constant and variable forces) is very small (about 1%). Thus, for the minimization of the static loads it is better to use constant force. The constant force is easier to create than the variable force.

Minimization of the torques due to the dynamic loads (inertia and gravitational forces).

The expressions for the input torques are similar to those in the previous case:

F A B M M i i i dyn i dyn ibal 3 , 3    (19)
The condition for the minimization of the torques at the p th calculated position is formulated as:

  , ,..., 1 , min 3 1 2 3 , 3 N p F A B M p F i i i i dyn i p         (20)
where

dyn i M
is the initial torque of the actuator i, N is the number of calculated positions of the simulation, i = 1, 2, 3 is the number of the actuator.

   N p A B M A B F i i i i i dyn i i i i p ,..., 1 , 3 1 2 3 , 3 3 1 3 , 3          (21)
Table 3 presents the maximum values of the torques for three examined cases. It should be noted that in this case also, the increase in the velocity and acceleration leads to the reduction of the efficiency of the minimization.

We would like to draw attention to the fact that the added force F is always vertical and cannot compensate for all effects of the inertia forces along the X and Y axes. It should be noted that these simulations showed that the minimization of the input torques achieved by using a variable force is not very efficient. The difference between two examined cases with constant and variable forces is very small. Taking into account the difficulty involved in the practical realization of the variable force, we can conclude that for the suggested balancing mechanism it is enough to use the constant force.

Increase in the positioning accuracy of spatial parallel manipulators balanced by the suggested mechanism

It should be noted that most research papers devoted to the study of parallel manipulators deal with the mechanical structures with rigid links. So in this case, the position of the platform is considered to be perfectly parallel to the base and its coordinates are determined from the nominal values of the link lengths. But in reality, the errors due to the elastic deformations of the mechanical structure of the manipulator change the position and orientation of the platform. Our observation showed that the increase in the platform mass leads to increases in these errors. In this section, it will be shown that the suggested balancing mechanism has a positive influence on the improvement of the positioning accuracy of the parallel robot.

  T z y x z y x M M M F F F  F .
Thus, we have

X K F   . ( 22
)
ΔX K F F   ( 23
)
where F bal = [0, 0, F bal , 0, 0, 0] T . Fig. 11 shows the positioning errors caused by the elastic deformation of the robot structure with balancing mechanism (light gray). Corresponding author: Arakelian

Table 5 shows a comparative analysis of the maximum values of the positioning errors along the corresponding axis for the two cases. The reduction in the positioning and orientation errors is significant (from 86.8% to 97.5%). After assembling the prototype, its static balance was verified by placing it vertically and noting that the mechanism is in equilibrium in any of its configurations. In such a manner, the . The base of the balancing mechanism was then suspended from the fixed structure of the Delta robot and its end was connected to the moving platform (Fig. 12). In order to create a balancing force, a counterweight was used. It is obvious that for industrial applications it is better to use pneumatic cylinders or electric actuators with constant moment. However, the validation of the obtained results can also be achieved by a counterweight, which develops the same force as a pneumatic cylinder or electric actuator.

Experimental bench

The experimental bench (Fig. 13) is composed of the Delta robot with its control system, a computer to interact with the user and a dSPACE 1103 board. The sampling period is 1 ms (corresponding sampling frequency f e ). The Delta Robot is composed of three Parvex RX320E DC servo motors with the following main characteristics:

-rated speed: 3000 tr/min and maximum speed: 3900 tr/min ; -rated torque (in slow rotation) : 1.08 Nm ; -rated current: 7.8 A and instantaneous maximum current: 20 A ; -100000 encoder pulses per revolution (resolution: 0.0036 degree). Corresponding author: Arakelian To show the improvement made in an industrial system by the balancing mechanism, we have preserved the existing industrial control system and used its speed control entries to actuate the motors. The dSPACE 1103 realizes the interface between the PC and the servosystem. This board allows the control of our 3 motorized axes and has specific entries to which we have directly connected our 3 incremental encoders. To control the system the software Matlab/Simulink/RTI/ControlDesk have been used.

The robot is controlled by a classical PID law, already included in the industrial control system. We have observed the torque response of the system to a movement composed of a straight line with respect to Z axis and a circle in the space (Fig. 14). Corresponding author: Arakelian Fig. 14. Selected trajectory for experimental validation of torque minimization.

Experimental results and comparative analysis

The input torques measured for each actuator in We would like to draw attention to cases C and D. In case C, we have only compensated the load added on the robot platform, to obtain the same result as thus obtained when the robot is unloaded. In case D, we have taken into account the mass of the Delta Robot links, which gives the best result.

The measured input torques have been taken for two different speeds of the movement (maximal cartesian acceleration and maximal cartesian speed):

Experiment 1 (E1): 15% of the maximum capacity of the robot to neglect most of the dynamic effects (we consider it as the static mode of operation).

Experiment 2 (E2): 100% of the maximum capacity of the robot to observe the improvement for dynamic mode of operation. Corresponding author: Arakelian The obtained measurements confirm perfectly the theoretical results (Fig. 15):

-When balancing is carried out by taking into account only the load on the platform, the results are similar to those obtained for the unloaded robot (cases A and C). Corresponding author: Arakelian -When balancing is carried out by taking into account the load on the platform and loads of robot links, we obtain the lowest values for the input torques (cases D).

Tables 6 and7 show the reduction of the input torques for experiments 1 (E1) and 2 (E2). We can observe that improvement for actuators 1 and 2 in the quasi static movement is 77% and for actuator 3 is 59%. For the dynamic mode of operation the improvement for actuator 1 is 52%, for actuator 2 is 56% and for actuator 3 is 18%. Corresponding author: Arakelian

Gains for actuator 3 are quite different from the two others, because this one was less solicited by the given trajectory than the two others, i.e. for the given trajectory the load of the platform on actuator 3 was smaller (see Table 7, case B). So, it is natural that for this actuator we do not observe consequent improvement of its torque by the balancing mechanism.

The experimental validation of the suggested balancing approach showed that satisfactory results are achieved and the developed system is fully operational. The next step of the experimental validation is the estimation of the positioning errors for balanced and unbalanced robots. For this purpose a trajectory given by the following 9 points was chosen (table 8).

These points are uniformly distributed about a straight line of 800 mm length. To obtain this line physically, a sphere is used, which was moved along a rail (Fig. 16). The position of each point is measured by three dial gauges, which determine the center of the sphere. The Corresponding author: Arakelian purpose of these measurements is to obtain the positioning errors for unbalanced and balanced Delta robots. With regard to the measurement of other positioning and orientation errors, caused by the displacement of the platform, we observed that the frame of the robot is insufficiently rigid in a cross-section direction. Taking into account the important mass of the counterweight, it is strongly deformed and leads to significant distortions of the measured parameters.

Conclusion

In this paper, a new approach for balancing of spatial parallel manipulators has been presented. It involves connecting a secondary mechanical system to the initial robot, which It should be noted that the suggested balancing mechanism not only improves the positioning accuracy of the parallel robot, but also sharply reduces stress in its links and efforts in the joints. This system can also be used for operational safety of robotized medical Corresponding author: Arakelian devices because it can maintain the fixed position of the platform if the parallel robot actuators should accidentally stop.

The suggested balancing approach was demonstrated for the Delta robot but the designed mechanism can be applied to many spatial parallel robots with 3 to 6 degrees of freedom. This type of mechanism is expected to lead to designs that can carry a larger payload and/or reduce energy consumption.

Finally, it should be noted that the proposed balancing mechanism has been patented [START_REF] Baradat | Parallel robot including of loadcompensation system[END_REF] and additional information is available upon request. Corresponding author: Arakelian

List of figures

List of tables

Fig. 1 .

 1 Fig. 1. A Delta robot used in the SurgiScope ® , a robotized navigation tool-holder designed for neurosurgery and developed by the ISIS 2 company.

  2 ISIS : Intelligent Surgical Instruments & Systems (http://www.isis-robotics.com). Corresponding author: Arakelian

Fig. 2 .

 2 Fig. 2. Principle of balancing.

Fig. 3 .

 3 Fig. 3. Simplified scheme of the balancing mechanism.

Fig. 4 .

 4 Fig. 4. Delta robot with the balancing mechanism.

Fig. 4

 4 Fig. 4 shows the balancing mechanism, which is implemented in the structure of the Delta

Fig. 5 .

 5 Fig. 5. Stewart platform with implemented balancing system.

Fig. 7

 7 Fig.7shows the workspace with the torque of the actuator 1 for each position of the

Fig. 6 .

 6 Fig. 6. Gravitational forces for the leg i.

  Corresponding author: Arakelian

Fig. 7 .

 7 Fig. 7. Input torque 1 for unbalanced (left) and balanced (right) Delta robot.

Fig. 8 .

 8 Fig. 8. The output parameters for the selected trajectory.

Fig. 9

 9 Fig.9shows the variations of the input torque 1 for the unbalanced and balanced Delta

Fig. 9 .

 9 Fig. 9. Input torque 1 for unbalanced and balanced Delta robot.

F

  , we determine the external force for each position of the trajectory: Corresponding author: Arakelian



Fig. 10 shows

 10 Fig.10shows the variations of the input torques for the initial and optimized cases (for a =

Fig. 10 .

 10 Fig. 10. Input torque 1 for unbalanced and balanced Delta robot.

  Errors caused by the linear displacements of the platform along the X axis. (d) Errors caused by the rotation of the platform about the X axis. Corresponding author: Arakelian (b) Errors caused by the linear displacements of the platform along the Y axis (e) Errors caused by the rotation of the platform about the Y axis. (c) Errors caused by the linear displacements of the platform along the Z axis (f) Errors caused by the rotation of the platform about the Z axis.

Fig. 11 .

 11 Fig. 11. Errors caused by the linear displacements and the rotation of the platform due to the

Fig. 12 .

 12 Fig. 12. CAD model and prototype of the balancing mechanism implemented

Fig. 13 .

 13 Fig. 13. Experimental bench.

  different cases are denoted by A, B, C and D: A. unloaded robot; B. loaded robot (robot with the load of 690 N); C. load balanced robot (balancing by mechanism with force of 690 N); D. load and mechanical system balanced robot (balancing by mechanism with force of 880 N).

(Fig. 15 .

 15 Fig. 15. Experimental measures of input torques for three actuators of the Delta robot.

Fig. 16 .

 16 Fig. 16. Measuring of the positioning errors for given straight line trajectory.

  . The abscissa axis corresponds to the unloaded case. Then the Delta robot was loaded and the relative errors were measured (graph "unbalanced robot"). Finally the robot was balanced by the suggested mechanism and relative errors are shown for the "balanced robot". The average rate of the improvement in the relative positioning accuracy with respect to Z axis is 93.5%, which corresponds to the value obtained by the numerical simulations.

Fig. 17 .

 17 Fig. 17. Relative positioning errors with respect to z axis

  generates a vertical force applied to the platform of the manipulator. The suggested balancing mechanism is designed on the base of the multiloop pantograph linkage introduced between the robot base and the platform. The minimization of the input torques was carried out by constant and variable forces for static and dynamic modes of operation. It was shown that a significant reduction in input torques can be achieved by the suggested balancing mechanism: the reduction of the root-mean-square value of the input torque due to the gravitational forces is 99.5% and the maximum value is 92%. The positioning errors of the unbalanced and balanced parallel manipulators are provided. It was shown that the elastic deformations of the manipulator structure due to the payload, change the altitude and the inclination of the platform. A significant reduction in these errors is achieved by using the balancing mechanism (from 86.8% to 97.5%). The theoretical results obtained by numerical simulations were confirmed by experimental study carried out by means of the developed prototype mounted on the Delta robot.

Fig. 1 .

 1 Fig. 1. A Delta robot used in the SurgiScope ® , a robotized navigation tool-holder designed for

Fig. 2 .

 2 Fig. 2. Principle of balancing.

Fig. 3 .

 3 Fig. 3. Simplified scheme of the balancing mechanism.

Fig. 4 .

 4 Fig. 4. Delta robot with the balancing mechanism.

Fig. 5 .

 5 Fig. 5. Stewart platform with implemented balancing system.

Fig. 6 .

 6 Fig.6. Gravitational forces for the leg i.

Fig. 7 .

 7 Fig. 7. Input torque 1 for unbalanced (left) and balanced (right) Delta robot.

Fig. 8 .

 8 Fig. 8. The output parameters for the selected trajectory.

Fig. 9 .

 9 Fig. 9. Input torque 1 for unbalanced and balanced Delta robot.

Fig. 10 .

 10 Fig. 10. Input torque 1 for unbalanced and balanced Delta robot.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Errors caused by the linear displacements and the rotation of the platform due to the

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Experimental bench.

Fig. 15 .

 15 Fig. 15. Experimental measures of input torques for three actuators of the Delta robot.

Fig. 16 .

 16 Fig. 16. Measuring of the positioning errors for given straight line trajectory.

Fig. 17 .

 17 Fig. 17. Relative positioning errors with respect to z axis for unbalanced and balanced robots.

Table 2 .

 2 Input torques for unbalanced and balanced robots

	Maximum values of	Maximum value of the input torque 1 (N.m)
				Balancing force
	the acceleration and				Gain
		unbalanced	balanced	(N)
	velocity
	a =0.1m/s²
		645	57	956	91 %
	v = 0.26 m/s
	a =1.05m/s²
		652	110	897	83 %
	v = 0.79 m/s
	a =4.13 m/s²
		677	326	779	49 %
	v =1.57 m/s

Table 3 .

 3 Input torques for unbalanced and balanced robots

	Maximum values of the acceleration	Maximum value input torque 1 (N.m)
				Gain
	and velocity	Unbalanced	Balanced
	a =0.1m/s²
		645	40	94 %
	v = 0.26 m/s
	a =1.05m/s²
		652	96	85 %
	v = 0.79 m/s
	a =4.13 m/s²
		677	262	60 %
	v =1.57 m/s

  Corresponding author: Arakelian Static rigidity of the Delta robot is defined as the 66 symmetrical matrix K that maps

	generalized infinitesimal displacements	ΔX		  x	y 	z 		x		y		z	 T	of the platform
	to generalized external loads

Table 4

 4

	. Link parameters

Table 5

 5

	. Maximal absolute positioning

Table 6 .

 6 Input torques (E1: static mode of operation)

							Improvement gain 3 (%)
			A	B	C	D
							C	D
	Maximum	Actuator 1	306	882	324	199	63	77
	values of the	Actuator 2	217	653	208	145	68	77
	measured
		Actuator 3	211	449	221	180	50	59
	torques (Nm)

Table 7 .

 7 Input torques (E2: dynamic mode of operation)

							Improvement gain 3 (%)
			A	B	C	D
							C	D
	Maximum	Actuator 1	456	880	502	423	43	52
	values of the	Actuator 2	291	608	342	264	44	56
	measured
		Actuator 3	313	400	320	328	20	18
	torques (Nm)

Table 8 .

 8 Trajectory for experimental validation of the positioning accuracy improvement.

	Points	x (mm)	y (mm)	z (mm)
	1	400.0	0.0	-900.0
	2	325.5	41.7	-949.9
	3	249.9	84.4	-1000.2
	4	175.6	127.5	-1050.0
	5	100.3	169.9	-1100.1
	6	25.5	212.4	-1149.8
	7	-49.9	255.4	-1199.7
	8	-124.8	298.3	-1249.6
	9	-200.8	340.7	-1299.8

Table 1 .

 1 Balancing schemes for robotic systems

Table 2 .

 2 Input torques for unbalanced and balanced robots

Table 3 .

 3 Input torques for unbalanced and balanced robots

Table 4 .

 4 Link parameters

Table 5 .

 5 Maximal absolute positioning and orientation errors for unbalanced and balanced robots

Table 6 .

 6 Input torques (E1: static mode of operation)

Table 7 .

 7 Input torques (E2: dynamic mode of operation)

Table 8 .

 8 Trajectory for experimental validation of the positioning accuracy improvement.

This gain represents in percent the reduction of the torques compared to (-) case B.

Acknowledgments

The authors would like to thank the «Intelligent Surgical Instruments and Systems» Company for its financial support and Professor Jean Le Flecher for the advice during development of the prototype.


Sébastien Briot

Cédric Baradat

Sylvain Guégan

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

CONTRIBUTION TO THE MECHANICAL BEHAVIOR IMPROVEMENT OF THE ROBOTIC NAVIGATION DEVICE SURGISCOPE ®

This paper proposes a new solution to the problem of torque minimization of the medical device SurgiScope ® by connecting to the initial structure a secondary mechanical system, which generates a vertical constant force on the platform of the robotized device. The conditions for optimization are formulated by the minimization of the rootmean-square values of the input torques of the studied device. The positioning errors of the unbalanced and balanced robots are provided. A significant reduction of these errors is achieved by using the suggested balancing mechanism. The efficiency of the developed approach is illustrated by numerical simulations. Experimental validation of the obtained optimization

INTRODUCTION

The SurgiScope ® (Fig. 1) is a ceiling mounted robotized tool-holder device which belongs to the surgical navigation products family. It is developed by the ISIS company. Such a 1 ISIS: Intelligent Surgical Instruments & Systems (http://www.isis-robotics.com). mechanism is especially dedicated to microscope applications in neurosurgery, and allows all kinds of operations in every usual patient's positions. The mechanical system, which carries the microscope is based on a Delta structure [START_REF] Clavel | Device for movement and displacing of an element in space[END_REF].

The Delta robot (Fig. 2) is well known in the electronics, food and pharmaceutical sectors as a reliable system with fast execution of light-duty tasks. However, in recent years, more attention has been paid to the increasing number of possible industrial applications, such as manipulation of medical devices.

In the case of the SurgiScope ® , the platform with the microscope is very heavy (about 70 kg) and consequently the required input torques are increased. Moreover, with such load, the positioning errors of the platform due to elastic deformations becomes non negligible.

The compensation of the input torque is a well-known problem in the field of the mechanical engineering [START_REF] Angeles | The optimum synthesis of an elastic torque-compensating cam mechanism[END_REF][START_REF] Nishioka | Design of torque compensation cam using measured torque distribution[END_REF][START_REF] Guilan | A new method of torque compensation for high speed indexing cam mechanisms[END_REF]. One of the methods used to solve this problem is to create a camspring mechanism in which the spring absorbs energy from the system when torque is low, and releases energy to the system when the required torque is high. This approach is considered to be the most efficient, such that a secondary mechanism compensates exactly the load variations in the mechanical system due to the periodic torque. Such a solution is successfully applied on the cyclic mechanisms with prescribed input motion characteristics.

FIG. 1. SURGISCOPE ® , A ROBOTIZED NAVIGATION TOOL-HOLDER DESIGNED FOR NEUROSURGERY

DEVELOPED BY THE ISIS COMPANY.

FIG. 2. SCHEMATIC OF THE DELTA PARALLEL ROBOT FROM CLAVEL'S PATENT [1].

However, for many industrial systems, such as robotic systems, this solution is not applicable because a simple camspring mechanism cannot be adapted to the variation of the input torque due to the trajectory planning.

Another trend for the solution of this problem is the balancing of the robot mechanisms. In this case the static balancing is defined as the set of conditions under which the weight of the links of the mechanism does not produce any torque at the actuator for any configuration of the robot. This condition is also referred to as gravity compensation. Gravitycompensated serial and parallel manipulators have been designed using counterweights, springs and sometimes cams [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF][START_REF] Fattah | Gravity-balancing of classes of industrial robots[END_REF][START_REF] Arakelian | Balancing of hand-operated manipulators[END_REF][START_REF] Arakelian | Balanced manipulator[END_REF][START_REF] Wang | Static balancing of spatial four-degree-of-freedom parallel mechanisms[END_REF][START_REF] Ebert-Uphoff | Static balancing of spatial parallel platform mechanisms," -Revisited, TransASME[END_REF].

However, for the complex systems, as for example the Delta parallel robot (Fig. 2), it is not easy to find any application for such a solution. The introduction of counterweights or springs brings the basic modification of the initial structure, which is constructively inefficient.

This paper proposes a new approach for the minimization of the input torques of the Delta robot. It is carried out by a secondary mechanical system, which generates a vertical force applied to the platform of the Delta robot.

The organization of the paper is as follows: first the description of the balancing mechanism is presented. Then, the detailed mathematical model for input torques minimization is formulated and an analytical solution is provided. In the next section, the influence of the balancing mechanism on the positioning errors of the platform is discussed. Finally, the prototype of the designed balancing mechanism is described and the experimental validation of the obtained optimization is presented.

DESCRIPTION OF THE BALANCING MECHANISM

For the minimization of the torques of the Delta robot with 3-DOF (Fig. 2) we suggest adding a secondary mechanism between the base (1) and the platform [START_REF] Arakelian | Balanced manipulator[END_REF]. The aim of this approach is to create on the platform (8) a supplementary vertical force F, which allows the minimization of the input torques of the robot. The designed balancing system, which has the mentioned characteristics, is based on the copying properties of the pantograph linkage (Fig. 3).

FIG. 3. COPYING PROPERTIES OF THE PANTOGRAPH LINKAGE.

The input points A and B of the pantograph are located in the horizontal and vertical guides of the rotating stand. So the suggested system has three degrees of freedom: a rotation of the stand about the vertical axis and two translations along the guides. This allows the suggested system to be passive relatively to the Delta robot when the point C is connected with the platform.

Point B is also connected with an actuator which produces a vertical force. This vertical force F B is used for the balancing of the gravitational forces of the spatial parallel robot. It is obvious that the determination of the balancing force F B = k F takes into account the magnification factor of the pantograph (k = AC/AB = a/b).

Thus the position of point C is represented by vector P = [x, y, z] T and the passive motions of the pantograph are represented by q = [r, , Z] T (r and Z represent the horizontal and vertical positions of point A and B, respectively, and  the rotation of the pantograph around Z axis). The kinematic relations between P and q are the following: x = (1-k) r cos, y = (1-k) r sin and z = k Z. Differentiating these equations with respect to time, one obtains:

P J q   1   (1) where   T z y x      P (2)   T Z θ r      q (3)                k θ r k θ k θ r k θ k 0 0 0 cos ) 1 ( sin ) 1 ( 0 sin ) 1 ( cos ) 1 ( J (4)
It is easy to see that a configuration singularity of the mechanism exists when x = y = 0, i.e. when the points A, B and C are on a vertical line (Fig. 3). To execute a continuous motion passing through such line, it may be necessary to orient the plane of the pantograph linkage relative to the Z axis. Thus, it is evident that the balancing mechanism must be equipped by a complementary rotating actuator for its orientation in the case of singular trajectories. This complementary actuator may be in operation only for special cases.

FIG. 4. DELTA ROBOT WITH THE BALANCING MECHANISM.

Fig. 4 shows the balancing mechanism, which is implemented in the structure of the Delta robot. Multiloop pantograph linkage with several link lengths allows the reduction of the overall size of the balancing mechanism. The sizes of the pantograph links must be chosen in such a manner that they should not collide with the legs of the Delta robot.

The next part will deal with the input torques compensation of the Delta robot.

INPUT TORQUES MINIMIZATION

In neurosurgery applications, the Delta robot is used in static mode operation. Thus, the input torque of the n th actuator can be expressed as:

n n n n n M M M M M 4 3 2 1    
(5) where M 1n is the torque due to the gravitational forces of the arms (see Fig. 5), M 2n is the torque due to the parallelograms, M 3n is the torque due to the joints on points B i , M 4n is the torque due to the gravitational forces of the platform and medical device. For i, n =1, 2, 3 and j =1, 2, 3 M jn is equal to:

n i ji T ji jn z y x z y x M             3 1 ) , , ( ) , , ( G J (6)
and for n = 1, 2, 3 and j = 4, st n 4

Μ is equal to:   n T n z y x z y x M G J   ) , , ( ) , , ( 4 (7)
where J 1i is the Jacobian matrix between the point P 1i and the actuated variables  n , J 2i is the Jacobian matrix between the point P 2i and the actuated variables n , J 3i is the Jacobian matrix between the point B i and the actuated variables n , J is the general Jacobian matrix of the robot, between the point P and the actuated variables  n for i, n = 1, 2, 3, and G and G ji are the gravity forces (Fig. 5). The matrix J ji (j = 1, 2, 3) can be written as:

. ; ; ;

3 2 2 1 1                                     n n i i n i i n i i     OP J OB J OP J OP J (8)

FIG. 5. GRAVITY FORCES FOR THE LEG i.

It should be noted that the Delta robot used in the medical device SurgiScope ® is symmetrical and the values of the input torques for the actuators are also symmetrical with altered zones (rotations of ±120°). Thus, in the following numerical illustrations we will only present the input torque for one actuator.

The three input torques can be presented in function of force F by the following expression:

                                F M M M M M M T bal bal bal 0 0 3 2 1 3 2 1 J (9)
where M ibal is the input torque of actuator i after balancing (i = 1, 2, 3).

The condition for the minimization of the root-meansquare (RMS) value of the torques can be expressed as:

  F N p i p p p i p p p i N F z y x J z y x M min ) , , ( ) , , ( 1 3 1 2 ) 3 (               (10)
where M i is the initial torque of the actuator i, N is the number of calculated positions of the robot, J (3i) is the i th column of the third line of the matrix J, i = 1, 2, 3 is the number of the actuator and x p , y p , z p are the coordinates of the p th calculated position of the workspace.

For the minimization of the RMS value of the torques, it is necessary to minimize the sum:

                 N k i p p p i p p p i F z y x J z y x M 1 3 1 2 ) 3 ( ) , , ( ) , , ( (11)
For this purpose, we shall achieve the condition: 0 /     F , from which we determine the force:

                          N k i p p p i N k i p p p i p p p i z y x J z y x M z y x J F 1 3 1 2 ) 3 ( 1 3 1 ) 3 ( ) , , ( ) , , ( ) , , ( (12)
Numerical simulation. For the Delta robot of the SurgiScope ® , the parameters are the following (see Fig. 5): -l AiBi =0,75 m; -l BiCi = 0,95 m; -m 1i = 2,3 kg (mass of i th arm with center P 1i ); -m 2i = 5,2 kg (mass of i th parallelogram with center P 2i ); -m 3i = 3,1 kg (mass of the joint at point B i ); -m = 79 kg (mass of the platform, joints and medical device, with center in the point P); -l AiP1i = l AiBi /2 = 0,375 m; -l BiP2i = l BiCi /2 = 0,475 m.

Thus the value of the added force for the given parameters is F = 931 N.

FIG. 6. ABSOLUTE VALUE OF INPUT TORQUE 1 FOR UNBALANCED (LEFT) AND BALANCED (RIGHT) DELTA

ROBOT.

Fig. 6 shows the variations of the input torques for unbalanced and balanced Delta robot. Thus, the reduction of the RMS value of the input torque is 99.5 %. The reduction of the maximum value of the torque is 92%.

It should be noted that the torques can also be minimized by the application of a variable force [START_REF] Baradat | Design of a torque-minimizing mechanism for the Delta parallel robot[END_REF]. However, the numerical simulations showed that in this way non significant improvement can be achieved (about 1%). Thus, for the minimization of the input torques of the medical device SurgiScope ® , it is better to use a constant force taking into account that it is much easier to create than a variable force.

IMPROVEMENT OF THE POSITIONING ACCURACY

It should be noted that most research papers devoted to the study of parallel manipulators deal with the mechanical structures with rigid links. So in this case, the position of the platform is considered to be perfectly parallel to the base and its coordinates are determined from the nominal values of the link lengths. But in reality, the elastic deformations of the mechanical structure of the manipulator change the position and orientation of the platform. Our observation showed that the increase of the payload leads to increase in these errors.

In this section, it is shown that the suggested balancing mechanism has a good influence on the improvement of the positioning accuracy of the parallel robot Delta.

Influence of the balancing mechanism on the positioning errors.

Static rigidity of the Delta robot is defined as the 66 symmetrical matrix K that maps generalized infinitesimal displacements of the platform X = [x, y, y,  x ,  y ,  z ] T to generalized external loads

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

X K F   . ( 13
)

TABLE 1. LINKS PARAMETERS.

Link 4 (Fig. 2) Link 5 (Fig. 2) Cross-section area (m²) 1.12410

FIG. 7. ERRORS CAUSED BY THE LINEAR DISPLACEMENTS AND THE ROTATION OF THE PLATFORM DUE TO THE ELASTICITY OF LINKS FOR UNBALANCED (DARK GRAY) AND BALANCED (LIGHT GRAY) DELTA ROBOT CALCULATED FOR THE ALTITUDE z = -1 m.

With the link parameters given in table 1 and the payload equal to 70 kg, the positioning errors caused by the elastic deformation of the robot structure is represented in Fig. 7 (dark gray). When the balancing force (F bal = 931 N) is applied on the platform, the relationship Eq. ( 13) can be rewritten as:

bal bal ΔX K F F   ( 14
)
where F bal = [0, 0, F bal , 0, 0, 0] T . Fig. 7 shows the positioning errors caused by the elastic deformation of the robot structure with balancing mechanism (light gray).

Table 2 shows a comparative analysis of the maximum values of the positioning errors along the corresponding axis for the two cases. The reduction in the positioning and orientation errors is significant (from 86.8% to 97.5%). Thus, it was shown that the balancing mechanism allows improving the positioning accuracy of the Delta robot. However, it is also interesting to study the following problem: to find such a force F, which allows carrying out the best improvement of the positioning errors.

Let us consider this problem.

Eq. ( 14) can be rewritten as:

bal bal ΔX ΔX F S    ( 15
)
where

S = K -1
The condition for minimizing the errors on the position is:

     
                    (16)
what leads to : The obtained force, determined from Eq. ( 17), is F bal = 924 N. The calculated errors are given in table 3. The comparative analysis shows that the obtained reduction of the deformations is not significant. This can be explained by the small difference between the applied forces (the difference is 7 N).

                          
Thus, it should be noted that, by reduction of the input torques of the Delta robot, a quasi best improvement of its positioning accuracy is also achieved.

In the next section we present the developed prototype and the experimental validation of numerical simulations.

PROTOTYPE AND EXPERIMENTAL VALIDATION

A prototype has been designed and built for validation of the obtained results. It was implemented in the structure of the Delta robot of the SurgiScope ® provided by the I.S.I.S Company. To design the prototype, it was firstly necessary to find the optimal lengths of the multiloop pantograph linkage taking into account that it should not collide with the legs of the Delta robot. Then, the right stiffness characteristics of the links were found by evolution of the shapes and design concept of rods, as well as by successive optimization based on the finite element analysis.

FIG. 8. CAD MODEL AND PROTOTYPE OF THE BALANCING MECHANISM IMPLEMENTED IN THE STRUCTURE OF THE DELTA ROBOT.

After assembling the prototype, its static balancing was readily verified and the force for the balance of the developed multiloop pantograph linkage was found through experimentation (F B/PANTOGRAPH = 52 N). The base of the balancing mechanism was then suspended from the fixed structure of the Delta robot and its end was connected to the moving platform (Fig. 8). In order to create a balancing force, a counterweight was used. It is obvious that for industrial applications, better design solutions could be found, as pneumatic cylinder or electric actuator with constant moment. However, the validation of the obtained results can also be achieved by a counterweight, which develops the same force as a pneumatic cylinder or electric actuator.

Experimental results and comparative analysis

We have observed the torque response of the system to a quasistatic movement composed of a straight line with respect to Z axis and a circle in the space (Fig. 9). We would like to draw attention to cases C and D. In case C, we have only compensated the load added on the robot platform, to obtain the same result as thus obtained when the robot is unloaded. In case D, we have taken into account the mass of the Delta robot links, which gives the best result.

The obtained measures confirm perfectly the theoretical results (Fig. 10): -when balancing is carried out by taking into account only the load on the platform, the result are similar to those obtained for the unloaded robot (cases A and C).

-when balancing is carried out by taking into account the load on the platform and loads of robot links, we obtain the lowest values for the input torques (cases D).

Table 4 shows the reduction of the input torques. We can observe that improvement for actuators 1 and 2 in the quasistatic movement is 77% and for actuator 3 is 59%.

Gains for actuator 3 are quite different from the two others, because this one was less solicited by the given trajectory than the two other, i.e. for the given trajectory the load of the platform on actuator 3 was smaller. So, it is natural that for this actuator we do not observe consequent improvement of its torque by the balancing mechanism.

The experimental validation of the suggested balancing approach showed that satisfactory results are achieved and the developed system is fully operational.

The next step of the experimental validation is the estimation of the positioning errors for balanced and unbalanced robots. For this purpose a trajectory given by the following 9 points was chosen (table 5). These points are uniformly distributed about a straight line of 800 mm length. To obtain this line physically, a sphere is used, which was moved along a rail (Fig. 11). The position of each point is measured by three dial gauges, which determine the centre of the sphere. The purpose of these measurements is to obtain the positioning errors for unbalanced and balanced Delta robots.

FIG. 11. MEASURING OF THE POSITIONING ERRORS

FOR GIVEN STRAIGHT LINE TRAJECTORY.

FIG. 12. RELATIVE POSITIONING ERRORS WITH RESPECT TO Z AXIS FOR UNBALANCED AND BALANCED

ROBOTS.

The obtained results are shown in Fig. 12. The abscissa axis corresponds to the unloaded case. Then the Delta robot was loaded and the relative errors were measured (graph "unbalanced robot"). Finally the robot was balanced by the suggested mechanism and relative errors are shown for the "balanced robot". The average rate of the improvement of the relative positioning accuracy with respect to the vertical axis is 93.5%, which corresponds to the value obtained by the numerical simulations.

CONCLUSION

The paper provided a new study concerning the mechanical behavior improvement of the medical device SurgiScope ® . The aim of this study is the minimization of input torques of the Delta robot used for robotic navigation of the medical device SurgiScope ® . For this purpose a new approach for balancing of Delta robot has been developed. It involves connecting a secondary mechanical system to the initial robot, which generates a vertical force applied to the platform of the robot. The suggested balancing mechanism is designed on the base of the multiloop pantograph linkage introduced between the robot base and the platform.

It was shown, by simulation and confirmed by experimental study, that a significant reduction of input torques can be achieved by using the suggested balancing mechanism: the reduction of the root-mean-square value of the input torques due to the gravitational forces is 99.5%. Moreover, a significant reduction of the positioning errors is achieved by using the balancing mechanism (from 86.8% to 97.5%).

It should be noted that the suggested balancing mechanism not only improves the positioning accuracy of the parallel robot, but also sharply reduces stress in its links and efforts in the joints. This system can also be used for operational safety of robotized medical devices SurgiScope ® because it can maintain the fixed position of the platform if the parallel robot actuators should accidentally stop.

Finally, it should be noted that the designed balancing mechanism has been patented [START_REF] Baradat | Parallel robot including of load-compensation system[END_REF] and additional information is available upon request.

( a )

 a Errors caused by the linear displacements of the platform along the X axis. (d) Errors caused by the rotation of the platform about the X axis. (b) Errors caused by the linear displacements of the platform along the Y axis (e) Errors caused by the rotation of the platform about the Y axis. (c) Errors caused by the linear displacements of the platform along the Z axis (f) Errors caused by the rotation of the platform about the Z axis.



  are respectively the values of S ij , x g , y g and z g of the p th calculated point of the workspace. 0.099 0.077 0.37 0.29 0.01 Gain % 88.9 89.3 97 91.5 91.4 92.3

FIG. 9 .

 9 FIG. 9. SELECTED TRAJECTORY FOR EXPERIMENTAL VALIDATION OF TORQUE MINIMIZATION. The input torques measured for each actuator in different cases are denoted by A, B, C and D: A. unloaded robot; B. loaded robot (robot with the load of 690 N); C. load balanced robot (balancing by mechanism with force of 690 N); D. load and mechanical system balanced robot (balancing by mechanism with force of 880 N).

(a) Input torque 1 (b) Input torque 2 (c) Input torque 3 FIG. 10 .

 12310 FIG. 10. EXPERIMENTAL MEASURES OF INPUT TORQUES FOR THE THREE ACTUATORS OF THE DELTAROBOT.

TABLE 2 . MAXIMAL ABSOLUTE POSITIONING AND ORIENTATION ERRORS FOR UNBALANCED AND BALANCED ROBOTS.

 2

	Constant balancing force
		(F bal = 931 N)
	Maximal absolute	Position (mm)	Orientation (10 -3 rad.)
	positioning errors	x	y	z	 x  y  z
	Unbalanced robot	0.92 0.923 2.636 4.35 3.37 0.13
	Balanced robot	0.109 0.107 0.065 0.41 0.31 0.02
	Gain %	88.2 88.4 97.5 90.6 90.8 86.8

TABLE 4 . INPUT TORQUES.

 4

			Maximum values of the measured torques
				(N.m)
			Actuator 1	Actuator 2	Actuator 3
	A		306	217	211
	B		882	653	449
	C		324	208	221
	D		199	145	180
	Improvement	C	63	67	50
	gain 2 (%)	D	77	77	59

TABLE 5 . TRAJECTORY FOR EXPERIMENTAL VALIDATION OF THE POSITIONING ACCURACY IMPROVEMENT.

 5

	Points	x (mm)	y (mm)	z (mm)
	1	400.0	0.0	-900.0
	2	325.5	41.7	-949.9
	3	249.9	84.4	-1000.2
	4	175.6	127.5	-1050.0
	5	100.3	169.9	-1100.1
	6	25.5	212.4	-1149.8
	7	-49.9	255.4	-1199.7
	8	-124.8	298.3	-1249.6
	9	-200.8	340.7	-1299.8

This gain represents in percent the reduction of the torques compared to the case B.

ACKNOWLEDGMENTS

The authors thank the "Intelligent Surgical Instruments and Systems" Company for its financial support and Professor Jean Le Flecher for the advice during development of the prototype.

is illustrated by a prototype developed in the National


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

Sébastien Briot

Sylvain Guégan

Jean Le Flecher

G C G M Ea3913

Design and Prototyping of New 4, 5 and 6 Degrees of Freedom Parallel Manipulators Based on the Copying Properties of the Pantograph Linkage

In this paper, a new family of parallel manipulators called PAMINSA 1 is proposed. The particularity of these manipulators is the decoupling of displacements in the horizontal plane from its translation along the vertical axis. The advantages of such an approach are discussed, and a prototype is presented. The positioning errors of the moving platform taking into account the elasticity of links are determined and the performances of such a design are shown.

INTRODUCTION

Mechanisms known as parallel manipulators are defined in the terminology for the mechanism and machine science [START_REF]Terminology for the mechanism and machine science[END_REF] as the manipulator that controls the motion of its end-effector by means of at least two kinematic chains going from the end-effector towards the frame. Such a mechanical architecture divides the manipulated load between the several legs of the system and, as a result, each kinematic chain carries only a fraction of the total load. Such a structural architecture allows for the reduction of the mass of the movable links and for the use of less powerful actuators. However, in spite of these advantages, there are some drawbacks, for example, a limited workspace or a high coupling between the different kinematic chains. The non-linearity of the static and dynamic models of the parallel manipulators is not attractive for industrial applications. That's why in the last few years new structures of parallel architectures have been developed for the decoupling of the kinematic and dynamic input/output relationships [START_REF] Kong | A Class of 3-DOF Translational Parallel Manipulator with Linear Input-Output Equations[END_REF][START_REF] Gosselin | A Fully Decoupled 3-DOF Translational Parallel Mechanism[END_REF][START_REF] Carricato | A Novel Fully Decoupled 2-DOF Parallel Wrist[END_REF][START_REF] Bouzgarrou | Rigidity Analysis of T3R1 Parallel Robot with Uncoupled Kinematics[END_REF]. The analysis of these schemes shows that, in most cases, fully decoupling is developed, i.e. the decoupling of the displacements about all the degrees of freedom of the platform. Despite rather-encouraging results, we would like to note that the fully-decoupled manipulators have drawbacks also, for example, a lack of stiffness or the increasing of the joint number. It is obvious that it is not easy to solve the problem of the full decoupling of the movements and to conserve the principal advantages of the parallel structures. That's why we tried to find a compromise 1 PAMINSA: Parallel Manipulator of the I.N.S.A. between the decoupling of the movements and the architectural particularity of the parallel structures.

A new approach to the problem of the design of decoupled parallel manipulators. An energetic analysis shows that the work of gravity applied on a point moving in the horizontal plane is equal to zero (the gravitational forces are always perpendicular to the displacements). But the work of the same force applied on a point moving along the vertical axis is other than zero (the gravitational forces are parallel to the displacements). This phenomenon is used in the design of the hand operated manipulators [START_REF] Arakelian | The history of the creation and development of hand-operated balanced manipulators[END_REF], in which the horizontal displacements of the payload are carried out manually and the vertical displacements are actuated. To design a new parallel architecture, we have applied this principle. In our previous works [START_REF] Briot | PAMINSA : un nouveau manipulateur d'architecture parallèle aux mouvements découplés[END_REF][START_REF] Arakelian | A New Decoupled Parallel Manipulator with Four Degrees of Freedom[END_REF][START_REF] Arakelian | Static and Dynamic Analysis of the PAMINSA[END_REF] the study of a new 4-dof parallel manipulator with decoupled motions was developed.

In this paper we generalize the adopted approach for the family of such manipulators from 4 to 6 degrees of freedom, and we show the performances of the prototype of the PAMINSA with 4-degrees of freedom.

THE FAMILY OF PAMINSA MANIPULATORS

The family of the PAMINSA manipulators is based on the copying properties of the pantograph linkage, which represents the kinematic chain between the fixed base of the manipulator and its end-effector. Thus each leg of the suggested manipulator is a pantograph linkage with two input parameters. This makes it possible to decouple the displacements between the horizontal plane and the vertical axis. In this manner, the displacements of the platform in the workspace are obtained by controlling the displacements of the input points of each leg. Figure 1 (a) and (b) represent two different versions of the legs of the PAMINSA. For the represented cases, the input points are A i and B i and the output point is C i . If the point A i is displaced horizontally, we obtain a horizontal displacement of the point C i . If the point B i is displaced vertically, we obtain a vertical displacement of the point C i . The application of the pantograph linkage allows the development of a new family of manipulators with 4, 5 or 6 degrees of freedom. It is possible to have different architectures for the same degree of freedom, differentiated by the number of legs between the base and the platform of the manipulator, as well as its actuated joints. In this paper we have only presented the manipulators with 3 legs and we note PAMINSA-iDjL a PAMINSA with i degrees of freedom and j legs. It should be noted that it is possible to create PAMINSA manipulators with 2 legs. These types of PAMINSA are presented in our previous work [START_REF] Arakelian | A New Decoupled Parallel Manipulator with Four Degrees of Freedom[END_REF]. Table 1 shows different architectures of the PAMINSA with three legs. For the PAMINSA-4D3L, the points A i are actuated on the horizontal plane by 3 motors M i which control directly the rotation of the revolute joints R i . The second possible version of this manipulator is the control of the translations in the prismatic pairs S i . For both versions the vertical displacements of the point B i are controlled by only one actuator M V . Thus, the motors M i control the displacement of the platform on the horizontal plane and the motor M V controls the displacement towards the vertical axis.

For the PAMINSA-5D3L, the rotations of the three legs allow the displacements of the platform in the horizontal plane. Points B i of two of the three legs are linked to the same actuator and the third leg has an independent vertical actuation. The vertical translations are controlled by the simultaneous vertical displacements of points B i of the 3 legs. The orientation about two axes is obtained by the vertical displacements of the point B i of the legs.

The PAMINSA-6D3L is a manipulator with 6 degrees of freedom composed of 3 legs. The rotations of the three legs allow the displacements of the platform on the horizontal plane. The point B i of each leg is controlled by an independent actuator. The vertical translations and the orientation of the platform are obtained as previously. However, in this case the platform can be oriented about three axes.

Advantages and possible applications of the PAMINSA.

The obvious advantages of the suggested manipulator architectures are the following:

-The decoupling of the control powers in two parts, making possible to raise an important payload to a fixed altitude by powerful actuators and then to displace it on the horizontal plane by less powerful actuators; -A great accuracy in the horizontal positioning because the payload can be locked in the horizontal plane by mechanical architecture of the manipulator, as well as the use of the less powerful actuators for the generation of motions on the horizontal plane (it is obvious that the less powerful actuators with a small load are more accurate then the powerful actuators). -The simplification of the vertical control based on linear input/output relationships (note that the vertical displacement of the platform is a copy of the linear displacement of the actuator multiplied by a magnification factor). Each kind of PAMINSA has its advantages and can be used differently. For example, the PAMINSA with 4 degrees of freedom allows improving the positioning accuracy about the vertical axis because the structure is mechanically locked during the displacement on the horizontal plane. We estimate in the following sections of this paper the positioning errors for a fixed altitude due to the structure deformations. Moreover, it allows improving the positioning accuracy on the horizontal plane along the x and y axes, because the loads due to the gravitational forces of the payload on the actuators M i (i=1, 2, 3) are insignificant (they are equal to zero after the balancing of the pantograph linkages, see [START_REF] Arakelian | Static and Dynamic Analysis of the PAMINSA[END_REF]). Such a design allows the fixation of an important load in a given altitude, then, positioning it on the horizontal plane by displacements towards two axes and the rotation about the vertical axis. The PAMINSA with 5 degrees of freedom allows the carrying out of all previous displacements with an inclination angle ψ of the platform (fig. 2). The angle of the inclination ψ can be defined as an angle between the normal N pl to the platform and the normal N of the plane xOy. Thus it is possible to move the platform on the horizontal plane with any inclination relative to the horizontal plane. In this case the inclination is defined by the rotation of the point C 1 about the line C 2 C 3 .

The PAMINSA with 6 degrees of freedom allows for any orientation of the platform about the z axis and the displacements of the platform on the horizontal plane. The angle of the inclination ψ can be defined as for the previous case but it can have any given value (i.e. any rotation about C 1 , C 2 and C 3 is possible). We would like to note that for the whole versions of the PAMINSA, there is a decoupling between the displacements on the horizontal plane and the displacements towards the vertical axis. The authors believe that the proposed manipulators could be used in industrial applications for the manipulation of heavy equipment with great positioning accuracy. Various fields are possible depending on the type of industrial application. The possible application of the PAMINSA could also be micromechanics for high-accuracy tasks. Indeed, thanks to the magnification factor of the pantograph linkage, a great displacement of the vertical actuators M v can be transformed into the small displacements of the platform.

THE PROTOTYPE OF THE PAMINSA-4D3L

We have developed in the I.N.S.A. of Rennes a prototype of PAMINSA-4D3L (Fig. 3). A short description of the developed manipulator is presented below. This prototype is based on the scheme presented in table 1. The displacements on the horizontal plane are obtained by actuating the revolute joint R i (Harmonic Drive motors connected with the legs by means of the toothed-belt transmission, Fig. 4). The actuator M v which controls the linear displacement of the vertical axes of the manipulator's legs is a linear actuator and, for the developed prototype, the PARVEX motor system was chosen.

The pantograph linkage has been carried out with double rods in order to improve the stiffness of the mechanical system. It is obvious that for an industrial application it is attractive to create a pantograph with single links and more rigid.

Error analysis due to the elastic deformations of the mechanical structure.

The stiffness of the developed prototype is studied taking into account the elasticity of the links of the pantograph linkages (the geometry and mass distribution parameters of the links are listed in Table 2). Two cases were examined on the CAD model (Fig. 5): the errors due to the deformations of the manipulator without any payload and with a load of 20 kg. Figure 6 shows the positioning errors due to the deformations of the manipulator in the first case. 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. However, it is important to note that the absolute errors along the vertical axis are rather small (z max = -0.25 mm and z min = -0.27 mm). The difference between these two values is less than 20 µm. Thus, we can note that the suggested manipulator allows moving of the platform on horizontal plane with great accuracy. It should be also noted that the positioning errors do not depend on the elasticity of actuator systems. The gravitational forces are also vertical and do not have any action on the rotating actuators. With a payload of 20 kg applied on the platform, the absolute variations of the positions along the vertical axis are z max = -1.22 mm and z min = -1.36 mm (Fig. 7). The difference between these two values is less than 140 µm, which is small, taking into account that the pantograph links are hollow tubes with a thickness of 1.5 mm. It is obvious that positioning errors for the manipulator can be reduced with high stiffness links.

Singular configurations.

The decoupling between the displacement on the horizontal plane and the displacement along the vertical axis allows the decoupling of the singularity analysis into parallel and serial singular configurations. Only the singular configurations of the planar projection of the suggested manipulator are represented below. It represents a planar 3-RPR parallel manipulator (Fig. 8).

The singular configurations of the planar 3-RPR parallel manipulator [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] appear if ρ 1 or ρ 2 or ρ 3 is equal to 0, where

i i i C O = ρ
represents the projection of the vector B i C i on the horizontal plane. In this case, the rotation of one of the legs of the manipulator does not allow for the platform displacement on horizontal plane. Another kind of singular configurations appears when the orientation angle of the platform is equal to:

) / arccos( b n R R ± = φ
or if the platform centre is located on a circle centred on point O with radius:

φ cos 2 2 2 n b b n R R R R R - + = ,
where R b is the base radius and R n is the platform radius. 9. This is the total workspace in which there are singular configurations. 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. Thus the workspace without the singular configurations is smaller. In order to eliminate the singular configurations, we think that it is reasonable to add two complementary linear actuators. It is important to note that these linear actuators will also be less powerful taking into account that the loads on the actuators which generate the displacements on the horizontal plane are very small (quasi-zero). Another trend, which can be interesting for the increase of the manipulator workspace without the singular configurations, is the optimal location of the frame's leg joints. In the developed prototype the legs are located symmetrically and this assemblage of the legs is not efficient. Such a disposition of legs was selected for further study of the singularity in dynamics. It seems that there can be a more efficient assemblage with an asymmetric location of legs, for example, one leg on one side and two legs on the other side of the platform. The space limitation of this paper did not permit the discussion of the optimal disposition of legs of the PAMINSA manipulators. In the future, we will make all possible efforts to provide a more detailed discussion regarding this problem.

CONCLUSIONS

In this paper, we have proposed a new family of parallel manipulators, in which the movements on the horizontal plane and the translations along the vertical axis are decoupled. Three types of such a decoupled manipulator with 4, 5 and 6 degrees of freedom are proposed. They are called PAMINSA and are divided into three groups. Each group has its advantages, which are discussed. The suggested structural architecture allows the application of the linear control of the vertical displacements and the cancellation of the loads on the actuators due to the gravitational forces during the displacements of the platform on the horizontal plane. The obtained results have shown that such design ensures a great absolute accuracy of the manipulator towards the vertical axis of the platform. Finally, it should be noted that the patent concerning of the PAMINSA manipulators is pending and additional information is available upon request.

Fig. 1 .

 1 Fig. 1. Two different versions of the legs of PAMINSA manipulators.

Fig. 2 .

 2 Fig.2. The angle of the inclination ψ of the platform.

Fig. 3 .

 3 Fig. 3. The prototype of PAMINSA-4D3L developed in the I.N.S.A. of Rennes.

Fig. 4 .

 4 Fig. 4. Actuation system of each leg.

Fig. 5 .Fig. 6 .

 56 Fig. 5. The CAD model of the PAMINSA-4D3L.

Fig. 7 .

 7 Fig. 7. Absolute positioning errors of the platform along the z axis with load of 20 kg (at altitude z=-0.6 m and with platform orientation φ =0°).

Fig. 8 .

 8 Fig. 8. Projection of the PAMINSA-4D3L on the horizontal plane.Workspace.The designed prototype has the following characteristics:-Platform radius R n = 100 mm, -Base radius R b = 350 mm, -Length D i C i = 630 mm, -Length A i D i = 420 mm, -Magnification factor of the pantograph linkage k= 3, -Point A i can move on the prismatic joint from a distance to the vertical axis of the pantograph between 50 mm and 300 mm, -The angle between the links A i D i et D i C i of the pantograph linkage is defined from 30° to 150°. The workspace of the developed prototype is shown in Fig.9. This is the total workspace in which there are singular configurations. 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. Thus the workspace without the singular configurations is smaller. In order to eliminate the singular configurations, we think that it is reasonable to add two complementary linear actuators. It is important to note that these linear actuators will also be less powerful taking into account that the loads on the actuators which generate the displacements on the horizontal plane are very small (quasi-zero). Another trend, which can be interesting for the increase

Fig. 9 .

 9 Fig. 9. Workspace of the developed prototype for a fixed orientation angle of the platform equal to 0°.

3 ( 1 )( 2 )

 312 Translations and 1 Rotation Type of connection between the platform and the legs: Universal (Cardan) joints Actuated joints : R1, R2, R3 et M v or S1, S2, S3 et M v PAMINSA-5D3L 3D view Projection on the horizontal plane 2 DOF: 3 Translations and 2 Rotations Type of connection between the platform and the legs: Spherical pairs Actuated joints : R1, R2, R3 and M v1 , M v2 or S1, S2, S3 and M v1 , M v2 PAMINSA-6D3L 3D view Projection on the horizontal plane 2 DOF: 3 Translations and 3 Rotations Type of connection between the platform and the legs: Spherical pairs Actuated joints : R1, R2, R3 and M v1 , M v2 , M v3 or S1, S2, S3 and M v1 , M v2 , M v3 Table 2. Dimensions and characteristics of the prototype's links Material: AU4G, Characteristics: E = 74000 MPa, ν = 0.33, ρ = 2800 kg/m 3 . Material: Steel, Characteristics: E = 210000 MPa, ν = 0.28, ρ = 7850 kg/m 3 . The masses of the joints: m Ai = 0.305 kg, m Bi = 0.338 kg, m Ci = 0.233 kg, m Di = 0.259 kg m Ei = 0.262 kg, m Fi = 0.28 kg, m Gi = 0.214 kg

Table 1 .

 1 PAMINSA manipulators

The projection of the platform on the horizontal plane changes with the variation of its orientation.


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

Sylvain Guégan

Sébastien Briot

G C G M Ea3913

STATIC AND DYNAMIC ANALYSIS OF THE PAMINSA (Parallel Manipulator of the I.N.S.A.)

Keywords: Parallel manipulator, decoupled motions, static optimization, dynamic modeling

In this paper we present an analytical approach for the static and dynamic analysis of the PAMINSA 1 , a new 4 degrees of freedom parallel manipulator that has been designed at the I.N.S.A. 2 in Rennes. On the base of the developed static model, the input torques due to the static loads are reduced by means of the optimum redistribution of the moving link masses. The analytical dynamic modeling of the PAMINSA by means of Lagrange equations is achieved. A numerical example and a comparison between the suggested analytical model and an ADAMS software simulation are presented.

INTRODUCTION

The complex nonlinear dynamics appears to be one of the most important parallel manipulator characteristics. Even in the static model, the expression of the torques (or forces) applied to the actuators due to the weight of the platform and links, are nonlinear. Driving torques on parallel manipulators are highly nonlinear functions of the position, velocity and acceleration of the mechanical actuator links. It should be noted that there are algorithms to regulate the problems of non-linearity (static or dynamic) and to ensure an efficient control and an acceptable computation cost. However, the simplification of the manipulator mechanical model is desirable and a mechanical system with a linear input/output relation is more appealing for industrial applications.

In the recent years, the decoupling of motions of parallel manipulators has attracted researchers' attention and different structures have been proposed [START_REF] Kong | A class of 3-dof translational parallel manipulator with linear input/output equations[END_REF][START_REF] Gosselin | A fully decoupled 3-dof translational parallel mechanism[END_REF][START_REF] Carricato | A novel fully decoupled 2-dof parallel wrist[END_REF][START_REF] Carricato | On the topological and geometrical synthesis and classification of translational parallel mechanisms[END_REF][START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot with uncoupled kinematics[END_REF][START_REF] Patarinski | Analysis and design of position/orientation decoupled parallel manipulator[END_REF][START_REF] Patarinski | Position/orientation decoupled parallel manipulators[END_REF][START_REF] Lallemand | The 6-dof 2-Delta parallel manipulator[END_REF]. Previous works on this problem may be arranged into two principal groups: (a) fully decoupled parallel structures in which the architecture of the manipulator is such that its input/output equations are linear and fully decoupled [START_REF] Kong | A class of 3-dof translational parallel manipulator with linear input/output equations[END_REF][START_REF] Gosselin | A fully decoupled 3-dof translational parallel mechanism[END_REF][START_REF] Carricato | A novel fully decoupled 2-dof parallel wrist[END_REF][START_REF] Carricato | On the topological and geometrical synthesis and classification of translational parallel mechanisms[END_REF][START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot with uncoupled kinematics[END_REF]; (b) position/orientation decoupled manipulators in which the end-effector's position is independent of the orientation [START_REF] Patarinski | Analysis and design of position/orientation decoupled parallel manipulator[END_REF][START_REF] Patarinski | Position/orientation decoupled parallel manipulators[END_REF][START_REF] Lallemand | The 6-dof 2-Delta parallel manipulator[END_REF].

Another trend of the kinematic decoupling is proposed in the design of the PAMINSA 3 . It consists in 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 3 k and 8 k , and an output point 5 k (k = 1, 2, 3). Each input point 8 k 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, ) 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 [START_REF] Arakelian | The history of the creation and development of hand-operated balanced manipulators[END_REF][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 their work due to the gravity of the manipulated object is equal to zero. It appears to us that the proposed manipulator could be used in industrial applications for the manipulation of heavy equipments with great positioning accuracy. Various fields are possible depending on the type of the industrial application.

This contribution deals with the static and dynamic analysis of the PAMINSA. The obtained results will be used for the optimization of the dynamic behaviour of a prototype, which is currently being developed at the I.N.S.A.

STATIC ANALYSIS AND INPUT TORQUES OPTIMIZATION

The torques (or forces) Q j applied to the actuators M j (j = 1, 2, 3, 4) due to the force of gravity of links, joints and platform of the studied manipulator can be expressed as:

) 4 , 3 , 2 , 1 ( , 3 1 1                j Q Q Q k n i st ikj st plj st j , ( 1
)
where st ikj Q is the load applied to the actuator j due to the gravity of the i-th link or bearings of the k-th leg (k = 1, 2, 3),

st plj
Q is the load applied to the actuator j due to the gravity of the platform. These loads can be represented in the form:

j k ik T ik st ikj z y x z y x Q               3 1 ) , , , ( ) , , , ( G J (2)   j pl T plj Q G J   , ( 3
)
where J ik is the Jacobian matrix between the point P ik and the actuated variables q j , P ik is the center of masses of the i-th link, J is the general Jacobian matrix of the robot, between the point P and the actuated variables q j , P is the center of masses of the platform, G and G ik are respectively the forces of gravity of the platform and the links (or bearings).

It is easy to see that:

  j pl T plj Q G J   =0, for j = 1, 2, 3. (4)
i.e. the input torques of the rotating actuators due to the gravity of the platform are cancelled because the gravitational forces are always perpendicular to the displacements (the platform carry out the displacements in the horizontal plane). Fig. 3 shows the redistribution of the actuator input torques for the whole workspace of the PAMINSA (for = /3, where  is the rotation angle of the platform about the z axis). The values of the input torques are differentiated by the different colors.

It should be noted that the workspace of the PAMINSA is symmetrical and the maximum values of the input torques are the same for all actuators but they are situated in different zones. Thus, the input torques of the rotating actuators are the following:

), 3 , 2 , 1 ( , ) , , , ( 3 1 1                j z y x Q Q k n i st ikj st j (5)
where

j k ik T ik st ikj z y x y x Q               3 1 ) , , , ( ) , , ( G J . ( 6
)
In the work [START_REF] Arakelian | Equilibrage des manipulateurs manuels[END_REF], it was shown that the input torques st j Q of the pantograph linkage can be cancelled by optimal redistribution of movable masses.

Thus, in the same way as in the work [START_REF] Arakelian | Equilibrage des manipulateurs manuels[END_REF] we can obtain: [START_REF] Patarinski | Position/orientation decoupled parallel manipulators[END_REF] and consequently

), 3 , 2 , 1 ( , 0   j Q st plj
), 3 , 2 , 1 ( , 0   j Q st j (8)
In conclusion one can note the input torques of the rotating actuators M 1 , M 2 and M 3 due to the gravity are completely eliminated and the displacements of the platform in the horizontal plane may be realized without great efforts (the residual efforts are due to the resistance due to the friction in the joints and the errors due to the elasticity of the links).

DYNAMIC ANALYSIS

The analytical model for dynamics of the PAMINSA based on the Lagrange equations has been formulated to compute the input torques which are necessary to control a given trajectory of the movable platform. In order to simplify the mathematical model, the effects of link elasticity and the friction in the joints have been neglected. The pantograph linkages are assumed to be composed of rigid bodies connected by joints without inner clearance.

It should be noted that a CAD model of the studied manipulator has been developed and more real simulation could be carried out on the software ADAMS.

The studied manipulator has 4 degrees of freedom, so it is

natural to select the 4 joint variables   Z , , , 3 2 1

  

as the 4 generalized coordinates, and then evaluate a set of 4 Lagrange equations for these coordinates. Such equations would be the formulas for the unknown torques (or forces).

However, due to the complexity of the geometrical model, the evaluation of the Lagrange function and especially its derivatives with four coordinates only, is found to be extremely involved. So we use the same approach as in the study [START_REF] Miller | The lagrange-based model of delta-4 robot dynamics[END_REF] and we increase the number of the generalized coordinates.

In this studied case, it is better to choose the following 8 generalized coordinates:

    Z z y x q j , , , , , , , 3 2 1      , j =1, …, 8 (9)
The notation used to describe the pantograph linkages is included below: (fig. 2)

B

I

, with

) ( ) ( i ZZ i YY I I  ( 10
)

Potential Energy

The potential energy V of the manipulator can be expressed as follow:

    3 1 k leg pl k V V V (11)
where,

z g m V pl pl   
is the potential energy of the platform,

3 3 9 2 5 1 e Z M z M z M V p k p k p leg k       
is the potential energy of the leg k, g is the gravitational acceleration, the expression of z 5k and z 9k are given in Appendix 1 and the expression of M pi , (i = 1, 2, 3) and e 3 are given in Appendix 2.

Kinetic Energy

The kinetic energy T of the manipulator can be represented in the form:

    3 1 k leg pl k T T T , (12)
where,

  2 2 2 2 ) ( 2 1             pl pl pl I z y x m T
is the kinetic energy of the platform,

k k k rot trans leg T T T  
is the kinetic energy of the leg k,

k c k c c k k c k k k k c k k k c k c k k c trans M Z z M Z M z z M y y x x M z y x M z M y x M T k                                          (13) k rot T
is the kinetic energy of the rotating links.

Let us note that there are two types of rotations (Fig. 4):

rotation due to the actuators M j (j=1, 2, 3) (angle  k ), which is about of the vertical axis,

rotation due to the displacement of the pantograph in the linkage plane (angles  k and  k ).

Fig. 4. A scheme for representation of the rotations of the k-th pantograph linkage.

The kinetic energy of the rotating links can be written as:

)) ( cos ) ( sin ) ( cos ) ( sin (
k c k c k c k c c k k c k c rot M M M M M M M T k                         (14)
The expressions for M ci (i = 1, …, 13) are given in Appendix 3.

So the Lagrange function of the system is given by the formula V T L   and the Lagrange equations are the following:

                   4 1 d d i ij i j j j A Q q L q L t  . ( 15
)
where  i are the Lagrange multipliers (i=1, …, 4), q j are the generalized coordinates (j=1, …, 8), Q j are the input torques or forces.

Coefficients A ij are obtained by differentiating the closureloop equations of the manipulator with respect to the generalized coordinates. These equations are given by the expressions:

0 ) cos( ) ( ) sin( ) ( 2 5 1 5        k k k k k k k e y e x s   , k=1, 2, 3 (16)
0 2 5 4      C k C l Z k l z s . ( 17
)
The expressions for  k , e 1k and e 2k are given in Appendix 1.

The system of equations ( 15) is solved as follow: firstly the Lagrange multipliers must be obtained from the first four Lagrange equations (for z y x q j , , ,  

) and then the input torques/forces can be determined from the last four Lagrange equations:

15 1 1 1 5 d d A L L t Q                     ( 18
) 26 2 2 2 6 d d A L L t Q                     ( 19
)
37 3 3 3 7 d d A L L t Q                     (20) 48 4 8 d d A Z L Z L t Q                 (21)

NUMERICAL SIMULATIONS

In order to verify the obtained expressions for the dynamic model of the PAMINSA, we compared a numerical application of the analytical computation with the ADAMS software simulations. For this purpose a CAD model of the PAMINSA (Fig. 5) has been developed with the parameters given in Appendix 4.

The variation of the input torques/forces for the prescribed trajectory (Appendix 5 -Fig. 8 to 11) with 1m/s² maximal acceleration has been computed and compared with the ADAMS simulations. The variation of the input torque of the rotating actuator 1 and the input effort of the linear actuator 4 for the entirely analytic dynamic model and the ADAMS simulation are shown in Fig. 6 and Fig. 7. The values of the input torques are differentiated by the different colors. The blue continuous curve represents the results of the analytic model and the red dotted curve the ADAMS simulation.

The obtained results show that the examined functions are identical. The small differences between the analytic dynamic model and the ADAMS simulation are probably caused by numerical noise.

Thus, the analytical dynamic model of the PAMINSA is validated and can be implemented to improve the performance of the control by taking into account, partially or totally all the dynamic interaction torques.

CONCLUSIONS

Static and dynamic analysis of the new parallel manipulator with 4 degrees of freedom, called the PAMINSA has been presented. By using an optimal redistribution of masses of the pantograph linkages, the input torques of the rotating actuators of the PAMINSA are cancelled. An analytical model based on the Lagrange equations was formulated and a simplified approach for its solution was proposed. Results of numerical simulations have been presented to show the feasibility of the proposed analytical approach.

     , 6 / 5 2 2       , 2 / 3 3 3                                  
F B A H    , ) 2 /( ) ( E K D F     , C E D K     4 2 , ) 1 ( 2    B E , ) ( 2 4 A X B D     , 2 4 2 4 2 3 2 A X A X l C B       , ) /( 4 1 X k Z B   , ) 2 /( ) ( 4 2 1 2 4 2 1 2 3 2 4 X k Z X X l l A B B        , k z Z 5 1  , ) 1 /( 1 4    k X X and 2 2 5 2 1 5 1 ) ( ) ( k k k k e y e x X     . Appendix 2                                              7 , 4 , 3 , 2 7 , 1 3 2 4 5 1 2 2 i i i Bi Bi B i p k m k m m k m m g M                     2 2 ) 2 ( ) 1 ( 8 4 7 3 9 7 4 2 B B B B p m m k m m k k m k m m k g M 2 1 3 B p m g M   ,            2 2 1 2 2 3 B B B m m m l g e Appendix 3                                 2 8 2 2 7 4 2 2 3 2 8 2 7 5 2 4 1 )) 1 ( 2 ( )) 1 ( 2 ( ) 1 ( 4
)) 1 ( 2 ( ) 2 ( ) 1 ( )) 1 ( ( 2 1 k m k k k m m k k k m k m k k m m k m M B B B B c                                       

Fig. 1 .

 1 Fig. 1. 3D view of the PAMINSA.

Fig. 2 .

 2 Fig. 2. Kinematic chain of each leg. The obvious advantages of the PAMINSA are the following: a) the decoupling of the drive powers into two groups, which allows the lifting of heavy load to a given altitude with only one very powerful actuator and then, by using other less powerful actuators, its accurate positioning in the horizontal plane, i.e. it is possible to use two kinds of actuators; b) a simplification of the vertical control based on the linear input/output equation (vertical displacements of the platform and the linear actuator are copied by a magnification factor); c) the improvement of positioning accuracy along the vertical axis because the mechanical locking of the structure does

  a) Input torque of the actuator 1. b) Input torque of the actuator 2. c) Input torque of the actuator 3.

Fig. 3 .

 3 Fig. 3. Actuator input torques.

  ik the link between the joints i and i+1 on the leg k m i the mass of i-th joint m Bi the mass of the link B ik l Bi the length of the link B ik i I the inertia matrix of the link B ikWe consider that the links are perfect tubes, i.e.

Fig. 5 .

 5 Fig. 5. CAD model of the PAMINSA.

Fig. 6 .

 6 Fig. 6. Comparison of the Lagrange dynamic model and ADAMS simulation for the input torque of the actuator 1.

Fig. 7 .

 7 Fig. 7. Comparison of the Lagrange dynamic model and ADAMS simulation for the input force of the actuator 4.

Parallel Manipulator of the I.N.S.A.

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

Patent concerning the PAMINSA is pending and a prototype is currently being developed.