Keywords: parallel manipulators, singularity, controllability, dynamics

It is known that a parallel manipulator with a singular configuration can gain one or more degrees of freedom and become uncontrollable. That is it might not reproduce a stable motion under a prescribed trajectory. However, it is proved experimentally that there is possible passing through the singular zones. This was simulated and shown on numerical examples and illustrated on several parallel structures. In this paper, we determine the optimal dynamic conditions generating a stable motion inside the singular zones. The obtained results show that the general condition for passing through a singularity can be defined as follows: the endeffector of the parallel manipulator can pass through the singular positions without perturbation of motion if the wrench applied on the end-effector by the legs, and external efforts of the manipulator are orthogonal to the twist along the direction of the uncontrollable motion. This condition is obtained from the inverse dynamics and analytically demonstrated by the study of the Lagrangian of a general parallel manipulator. Numerical simulations are carried out using the software ADAMS and validated by experimental tests.

I. INTRODUCTION

arallel manipulators have experienced an increase in popularity in recent years due to their higher rate of acceleration, payload to weight ratio, stiffness and low effective inertia compared to those of serial manipulators. However they have some drawbacks, like a small workspace and special singular zones in it. Thus, in the presence of singular positions, the workspace of the parallel manipulators, which is less than that of serial manipulators, becomes even smaller and limits their functional performance. The studies of singularity have had different stages of development. The previous work on this problem is reported in a great number of publications and can be classified by different criteria. We arrange them in three major groups, distinguished by study of the singularity from kinematic, kinetostatic and dynamic points of view.

The physical interpretation of a singularity in kinematics refers to those configurations of parallel manipulators 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 Manuscript received September 12, 2007. S. Briot and V. Arakelian are with the National Institute of Applied Sciences (I.N.S.A.), Rennes, 35043 France (corresponding author: +33223238492; fax: +33223238726; e-mail: vigen.arakelyan@insarennes.fr). loses some degrees of freedom. In this case the singularity analysis can be carried out on the base of the properties of the Jacobian matrices of the mechanical structure (i.e. when the Jacobian matrices relating the input speeds and the output speeds become rank deficient [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Ma | Architecture singularities of parallel manipulators[END_REF][START_REF] Ottaviano | Singularity analysis of CaPaMan: a three-degree of freedom spatial parallel manipulator[END_REF][START_REF] Wen | Singularities in three-legged platformtype parallel mechanisms[END_REF]), by using Grassmann geometry [START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF] or screw theory [START_REF] Hunt | Special configurations of robot-arms via screw theory[END_REF][START_REF] Bonev | Singularity analysis of 3-DOF planar parallel mechanisms via Screw Theory[END_REF].

However, it was observed that close to a singular configuration, a parallel manipulator loses its stiffness and its quality of motion transmission, and as a result, its payload capability. Thus, there are not only positions but rather singularity zones which must be avoided; and an indicator of the quality of motion transmission close to the singular configurations of parallel manipulators must be defined. For this purpose, a kinetostatic approach has been applied for the evaluation of the quality of motion transmission.

The quality of motion transmission of parallel manipulators was successfully studied in [START_REF] Kim | Forward/inverse force transmission capability analyses of fully parallel manipulators[END_REF][START_REF] Lee | Workspace and forcemoment transmission of a variable arm type parallel manipulator[END_REF][START_REF] Kim | Force transmission analyses with dimensionally homogeneous jacobian matrices for parallel manipulators[END_REF][START_REF] Weiwei | Study on force transmission performance for planar parallel manipulator[END_REF]. The quality of motion of manipulators with three degrees of freedom has been evaluated by means of a kinetostatic indicator, which is similar to the pressure angle [START_REF] Alba-Gomez | Consistent kinetostatic indices for planar 3-DOF parallel manipulators, application to the optimal kinematic inversion[END_REF]. In [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF], the pressure angle was used as an indicator of the quality of motion transmission and the nature of the inaccessibility of singular zones by parallel manipulators are shown.

The study of singularities in parallel manipulators has revealed an interesting problem that concerns the path planning of parallel manipulators under the presence of singular positions, i.e. the motion feasibility in the neighborhood of singularities. In this case the dynamic conditions can be considered in the design process.

One of the most evident solutions for the stable motion generation in the neighborhood of singularities is to use redundant sensors and actuators [START_REF] Collins | The singularity analysis of an parallel hand controller for force-reflected teleoperation[END_REF][START_REF] Dasgupta | Force redundancy in parallel manipulators: theoretical and practical issues[END_REF][START_REF] Glazunov | Parallel manipulator control while intersecting singular zones[END_REF][START_REF] Alvan | On the control of the spatial parallel manipulators with several degrees of freedom[END_REF]. However, it is an expensive solution to the problem because of the additional actuators and the complicated control of the manipulator caused by actuation redundancy. Another approach concerns with motion planning to pass through singularity [START_REF] Bhattacharya | Comparison of an exact and approximate methode of singularity avoidance in platform type parallel manipulators[END_REF][START_REF] Dasgupta | Singularity-free path planning for the Steward platform manipulator[END_REF][START_REF] Perng | Inverse kinematic solutions for a fully parallel robot with singularity robustness[END_REF][START_REF] Ider | Inverse dynamics of parallel manipulators in the presence of drive singularities[END_REF][START_REF] Kevin | Path tracking of parallel manipulators in the presence of force singularity[END_REF][START_REF] Dash | Workspace generation and planning singularity-free path for parallel manipulators[END_REF][START_REF] Nenchev | Singularity-consistent path planning and control of parallel robot motion through instantaneousself-motion[END_REF][START_REF] Nenchev | Dynamic analysis of parallel manipulator under the singularity-consistent parameterization[END_REF], i.e. a parallel manipulator may track a path through singular poses if its velocity and acceleration are properly constrained. This is a promising path for the solution of this problem. However only a few research papers on this approach have addressed the path planning for obtaining a good tracking performance but they have not adequately addressed the physical interpretation of dynamic aspects.

In this paper, for the first time, the condition for passing through the singular positions is defined in general. It can be formulated as follows: "In the presence of a type 2 singularity, the platform of the parallel manipulator can pass through the singular positions without perturbation of

Sébastien Briot and Vigen Arakelian

On the Dynamic Properties and Optimum Control of Parallel

Manipulators in the Presence of Singularity P motion if the wrench applied on the platform by the legs and external forces is orthogonal to the direction of uncontrollable motion". In other terms, the condition is that the work of applied forces and moments on the platform along the uncontrollable motion is equal to zero. This condition is obtained from the inverse dynamics and analytically demonstrated by the study of the Lagrangian of a general parallel manipulator. The obtained results are illustrated by numerical simulations and validated by experimental tests. The paper is organized as follows. The next section presents theoretical aspects of the examined problem. Based on the Lagrangian formulation, the condition of force distribution is defined, that allows the passing of any parallel manipulator through the type 2 singular positions. In Section 3, an application of the obtained theoretical results is examined for a parallel manipulator with 4 degrees of freedom. In section 4, the numerical simulations carried out using the software ADAMS are validated by experimental tests II. OPTIMAL DYNAMIC CONDITIONS FOR PASSING THROUGH TYPE 2 SINGULARITY Let us consider a parallel manipulator of m links, n degrees of freedom and driven by n actuators.

The Lagrangian dynamic formulation for a parallel manipulator can be expressed as:

λ B q q T L L + ∂ ∂ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = & dt d τ , (1)
where, -τ is the vector of the input efforts; -L is the Lagrangian of the manipulator; -

T n q q q ] ,..., , [ 2 1
= q and T n q q q ] ,..., , [

2 1 & & & & = q
represent the vector of active joints variables and the active joints velocities, respectively; -

T z y x ] , , , , , [ θ ψ φ = x and T z y x ] , , , , , [ θ ψ φ = v
are trajectory parameters and their derivatives, respectively; x, y, z represent the position of the controlled point and φ, ψ and θ the rotation of the platform about three independent axes a φ , a ψ and a θ ); -λ is the Lagrange multipliers vector, which is related to the wrench applied on the platform by:

p W A T - = λ (2)
where, -A and B are two matrices relating the vectors v and q & according to q B Av & =

. They can be found by the closure equations with respect to time.

-W p is the wrench applied on the platform by the legs and external forces, which is defined as:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = p p p n f x v W L L dt d . ( 3
)
where f p is the force along the directions of the global frame and n p is the torque about the axes a φ , a ψ and a θ .

The term W p can be rewritten in the base frame using a transformation matrix D [START_REF] Merlet | Parallel Robots[END_REF]:

) ( 0 p R p W D W = (4) with p R W 0
is the expression of the wrench W p in the base frame, and

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = × × × × 3 3 3 3 3 3 3 3 R 0 0 I D (5)
where I 3×3 , 0 3×3 and R 3×3 are respectively the identity matrix, the zero matrix and the transformation matrix between axes a φ , a ψ and a θ and the base frame, which dimensions are 3×3. By substituting ( 5) into (1), one can obtain:

p R b W J W 0 T + = τ , q q W b ∂ ∂ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = L L & dt d ( 6
)
where

( ) B A J R 1 0 - =
is the Jacobian matrix between twist t of the platform (expressed in the base frame) and q & ,

D A A R = 0
is the expression of matrix A in the base frame. For any prescribed trajectory x(t), the values of vectors q & & , q & and q can be found using the inverse kinematics. Thus, taking into account that the manipulator is not in a type 1 singularity [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF], the terms W b and p R W 0 can be computed.

However, for a trajectory passing through a type 2 singularity, the determinant of matrix J tends to infinity. Numerically, the values of the efforts applied by the actuators become infinite. In practice, the manipulator either is locked in such a position of the end-effector or it generates an uncontrolled motion. That is the end-effector of the manipulator produces a motion, different to the prescribed trajectory.

It is known that a type 2 singularity appears when the determinant of matrix A R 0 vanishes, in other words, when at least two of its columns are linearly dependant [START_REF] Merlet | Parallel Robots[END_REF]. [START_REF] Hunt | Special configurations of robot-arms via screw theory[END_REF] represents the direction of the wrench applied on the platform by the i-th actuator of the mechanism [START_REF] Merlet | Parallel Robots[END_REF].

Let us rewrite the matrix

A R 0 as: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 6
R R R A R M K M O M M K K a a a a a a a a a (7) In this expression, vector R i = [a i1 , a i2 , …, a i6 ] (i = 1, …,

If the columns of matrix

A R 0
are linearly dependant, we can write:

∑ = = 6 1 0 j ij j R α , i = 1, …, 6 (8)
where R ij is the j-th component of the vector R i and α j are the coefficients, which in general can be functions of q p , p = 1, …, n). Thus, it is possible to show that the vector t s =

[α 1 , α 2 , …, α 6 ] T is the reciprocal twist to the wrenches R i .

This vector represents the direction of the free motion of the platform in a type 2 singularity.

Rewriting (8) in a vector form, we obtain:

∑ = = 6 1 j j j 0 N α , N j = [a 1j , a 2j , …, a nj ] T , j = 1, …, 6 (9),
where N j represents the j-th column of matrix A R 0 . By substituting ( 9) into (2), we have

j T j W = λ N , j = 1, …, 6 (10)
, where W j is the j-th row of vector p R W 0 .

Then, from ( 9) and ( 10) the following conditions are derived:

( ) ( ) ∑ ∑ = = = = 6 1 6 1 0 j j j j T j j W α α λ N ( 11
)
The right term of eq.( 11) corresponds to the scalar product of vectors t s and p R W 0 .

Thus, in the presence of a type 2 singularity, it is possible to satisfy conditions [START_REF] Weiwei | Study on force transmission performance for planar parallel manipulator[END_REF] if the wrench applied on the platform by the legs and external efforts are orthogonal to the direction of the uncontrollable motion (singular motion). Otherwise, the dynamic model is not consistent. Obviously, in the presence of a type 2 singularity, the displacement of the end-effector of the manipulator has to be planned to satisfy [START_REF] Weiwei | Study on force transmission performance for planar parallel manipulator[END_REF]. In the case of an inconsistent trajectory close to or through the singular positions, the required input efforts gain importance and a feasible motion is unattainable.

III. ILLUSTRATIVE EXAMPLE

In this section, the above discussed theory is illustrated by the example of a spatial parallel manipulator, PAMINSA, developed in the I.N.S.A. of Rennes, France [START_REF] V. Arakelian | Parallel robot comprising means for setting in motion a mobile element split in two separate subassemblies[END_REF]. Then, the results of numerical simulations carried out by the software ADAMS are tested on the built prototype.

The particularity of the PAMINSA, is in decoupling of the displacements of the platform in the horizontal plane from the translations along the vertical axis. The advantages of such an approach was discussed in [START_REF] Arakelian | Design and prototyping of new 4-, 5-and 6-DOF parallel manipulators based on the copying properties of the pantograph linkage[END_REF][START_REF] Arakelian | Singular positions of a PAMINSA parallel manipulator[END_REF] and the singularity analysis is discussed in [START_REF] Briot | On the Design of PAMINSA: a New Class of Parallel Manipulators with High-Load Carrying Capacity[END_REF][START_REF] Briot | Singularity analysis of PAMINSA manipulators[END_REF].

The previous studies have revealed that there are type 2 singularities in the workspace of the symmetrical architecture of PAMINSA. In this section, we study the possibility of passing through the singular positions by this manipulator.

Fig. 1 shows the PAMINSA with four degrees of freedom. Each leg of the manipulator is realized by a pantograph mechanism (Fig. 2) with two input points 3 i and 8 i , and an output point 5 i (i = 1, 2, 3). Each input point 8 i is connected to the rotating drive M i by means of a prismatic guide mounted on a rotating link. This kind of architecture allows for generation of motion in the horizontal plane by the use of rotating actuators M 1 , M 2 , M 3 , and the vertical translations by means of the linear actuator M v . Thus, the displacements (x, y, φ) of the platform in the horizontal plane xOy, that are translations along the x and y-axes and rotations about the z-axis, are independent of vertical the lengths PC i and OM' i (Fig. 3). In this case, the manipulator gains one infinitesimal rotation around one vertical axis. c) the platform is located in a circle defined by

φ cos 2 2 2 2 2 b pl b pl R R R R y x - + = + (15)
In this case, the manipulator gains one finite rotation about one vertical axis (Cardanic self motion) [START_REF] Chablat | Self motions of a special 3-RPR planar parallel robot[END_REF].

For cases (b) and (c), the direction of the unconstrained motion can be represented by the twist t s = [0, 0, 1, x W , y W , 0] T , where x W and y W correspond to the planar coordinates of the intersection point of the wrenches R i applied on the platform by three legs of the manipulator (Fig. 3). Let us now study the inverse dynamics of the PAMINSA. The potential energy V can be written as:

∑ = + = 3 1 i leg pl i V V V ( 16
)
where V pl is the potential energy of the platform and i leg V is the potential energy of the leg i (i = 1, 2, 3). By further considering that the coordinates of the all points of the pantograph linkages can be found as a linear combination of the coordinates of points 3 i , 5 i and 9 i , one can express the terms V pl and i leg V

as follows:

z g m V pl pl = (17) 4 3 9 2 5 1 v v v i v i v leg C q C z C z C V i + + + = (18)
Here, C vj (j = 1, 2, 3) are constant terms whose dimension is equivalent to a mass multiplied by the gravitational acceleration g, m pl is the mass of the platform with a payload, and z 5i and z 9i are the altitude of joints 5 i and 9 i . The expressions of the coordinates of joints 5 i and 9 i , as well as the expressions for C vj are complex, and their inclusion into this paper under the limited allowed length is not possible. An interested reader is referred to [START_REF] Briot | Analysis and optimization of a new family of parallel manipulators with decoupled motions[END_REF].

We consider that the links are perfect tubes. Therefore the tensor of inertia I j of the link B ji at the center of masses will be written as:

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ) ( ) ( ) ( 0 0 0 0 0 0 j ZZ j YY j XX j I I I I , with ) ( ) ( j ZZ j YY I I = (19)
Thus, the kinetic energy T of the manipulator can be represented as:

∑ = + = 3 1 i leg pl i T T T , (20)
where T pl is the kinetic energy of the platform, T legi is the kinetic energy of the leg i, as ( )

2 2 2 2 ) ( 2 1 φ & & & & pl pl pl I z y x m T + + + = ( 21
)
where I pl is the axial moment of inertia of the platform about the vertical axis. ) (

) ( ) ( i c v i c v c i i c i i i i c i i i c i c i i c trans q C q z C q C z z C y y x x C z y x C z C y x C T + + + + + + + + + + + = (23) i rot
T is the kinetic energy of the rotating links.

Note that there are two types of rotations (see, Fig. 1): -rotation due to the actuators M i (i = 1, 2, 3) (angle q i ), which is about the vertical axis, -rotations due to the displacement of the pantograph in the linkage plane (angles ζ i and ε i denoted as the angles between the direction of the passive slider and the links B 4i and B 3i respectively). Thus, the kinetic energy of the rotating links can be written as:

i c i c i c i c c i i c i c rot C C C C C q C C T i ε ε ς ς ε ζ + + + + + + = & & & ( 24
)
The expressions for C cj (j = 1, …, 13) are given in [START_REF] Briot | Analysis and optimization of a new family of parallel manipulators with decoupled motions[END_REF].

The input torques can be obtained from ( 6):

p b W J W T PAM + = τ (25),
where the terms J PAM , W b and W p are presented in [START_REF] Briot | On the Design of PAMINSA: a New Class of Parallel Manipulators with High-Load Carrying Capacity[END_REF].

The following parameters of manipulator's links are specified for the trajectory generation:

the radii of the circles circumscribed to the base and terms of the inertia matrices of the links of the pantograph linkages: The point P is desired to make a motion x(t) along a straight line between points P 0 (x 0, y 0 ) = P 0 (0, 0) and point P f (x f , y f ) = P f (0.3, 0) in t f = 2.4 s. However, the manipulator will pass through a type 2 singular position at point P s (x s , y s ) = (0.25, 0) (Fig. 4). In order to carry out a comparative analysis for the optimized and not optimized dynamic conditions for passing through type 2 singularity, it has been considered two cases. The first is such a movement on the given trajectory, which is calculated from condition [START_REF] Weiwei | Study on force transmission performance for planar parallel manipulator[END_REF], and the second is an arbitrary motion.

At first let us consider an optimized trajectory which allows satisfying the condition (11), i.e. the force W p should be perpendicular to the to the twist t s [0, 0, 1, 0, 0.1, 0] T defining the direction of the unconstrained motion. Developing the expression [START_REF] Weiwei | Study on force transmission performance for planar parallel manipulator[END_REF] for the PAMINSA at point P s , we obtain: Now considering that the end-effector of the manipulator moves along a straight line directed along the x-axis, we can note that

) ( s t y & = ) ( s t z & = ) ( s t y& & = ) ( s t z& & = ) ( s t φ & = 0 ) ( = s t φ & &
. Thus, the relationships, which satisfy the passing through of the singular positions, taking into account that the velocity and the acceleration of the platform in the initial and final positions are equal to zero, can be expressed by the following boundary conditions:

x(t 0 ) = x 0 , ( 27)

x(t f ) = x f , ( 28
) x(t s = 2s) = x s , ( 29
) 0 ) ( 0 = t x & , ( 30
) 0 ) ( = f t x & , (31) 0
) ( 0 = t x& & , ( 32
) 05 . 0 ) ( = = s s x t x & & m/s, ( 34
) 1.32583 ) ( - = = s s x t x m/s².
(35) In this case, a motion for passing of the platform through the singular position can be found from the following eighth order polynomial form: ( ) However, a trajectory obtained by (36) cannot be reproduced by the prototype because of the limited capability of drivers' deceleration. Therefore, the trajectory was divided into two parts, i.e., the first sixth order polynomial trajectory assumes the motion from an initial to the singular position (P 0 P s ) and the second sixth order polynomial trajectory-from singular to the final position (P s P f ). The core of the problem is same but it allows for generating motions for the prototype. Thus, the trajectory planning equations can be written as: Thus, the motion obtained from sixth order polynomial equations allows for passing through the singularity without perturbation, and the input efforts take on finite values (Fig. 5).

( ) ( )( ) 6
It can be seen that by the end of the motion there is an increase of the input efforts, caused by a quick deceleration to stop the manipulator before it reaches the workspace boundary.

It will be shown further that in the case of the motion generated by any trajectory planning without meeting the adopted boundary conditions ( 27) -( 35), the manipulator platform is not able to pass through the singular position. For demonstration purpose, the generation of motion between initial and final positions is carried out by a fifth order polynomial trajectory planning.

In this case, for y(t) = 0m, z(t) = -0.45m and φ(t) = 0°, the fifth order polynomial trajectory planning is the following:

( ) ( ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + = 5 4 3 0 0 6 15 10 f f f f t t t t t t x x x t x ( 39
)
The obtained input efforts are represented in Fig. 6.

It can be noted that for arbitrary trajectory while the manipulator passes through the singular configuration (for t s ≈ 1.8 s), the value of the input torques tend to infinity.

IV. EXPERIMENTAL VALIDATION OF OBTAINED RESULTS

For validating the results of the previous section, we have carried out experimental tests on the prototype of the PAMINSA with 4 DOF developed in the I.N.S.A. of Rennes (Fig. 7).

We have applied an arbitrary fifth order control law and observed the reproduction of motion during the displacement of the platform. The obtained trajectory is shown in Fig. 8 (dotted line).

The different positions are classified by time. For positions from (a) to (d), the platform moves towards the singular zone but yet it is outside of it. In this case, the reproduction of the real trajectory is similar to the desirable. At position (e), the manipulator enters the singular zone, which is close to the circle of the theoretical singular loci, and starts an uncontrollable motion. Thus, since the motion generation is carried out by non optimized dynamic parameters, the platform moves along an unplanned trajectory (see positions (e) -(h) in Fig. 8).

Next, we have implemented the sixth order control laws as it was shown in the previous section and observed the behavior of the platform during the displacement. The different positions are classified by time. During all these displacements, the manipulator retains its orientation and passes through the singular configuration without any perturbation.

V. CONCLUSION

The singularities limit the functional performances of parallel manipulators. The best known solutions to increase the size of the workspace of parallel mechanisms, based on the actuation redundancy, are expensive and cannot be widely adopted. Therefore, other means for passing through singular configurations need to be developed.

In this paper we have found the optimal dynamic conditions, for making the pass through the type 2 singular configurations possible. The general definition of the condition for passing through the singular position is formulated as follows: in the presence of type 2 singular configuration, the platform of a parallel manipulator can pass through the singular positions without perturbation of motion if the wrench applied on the platform by the legs and external efforts are orthogonal to the direction of the uncontrollable motion, or in other words, if the work of applied forces and moments on the platform along the uncontrollable motion is equal to zero. This condition has been verified by numerical simulations carried out with the software ADAMS and validated by experimental tests on the prototype of four degrees of freedom parallel manipulator PAMINSA.

We point out that for the case of non controllable external forces applied on the platform the proposed technique cannot be used. Therefore, the most prominent field of the industrial application is a "fast pick and place" manipulation, when the generation of motion is determined by input, gravitational and inertia forces.

Fig. 1 .

 1 Fig. 1. PAMINSA: Parallel Manipulator of the I.N.S.A. [27].

Fig. 2 .

 2 Fig. 2. Kinematic chain of each leg. translations z. This implies that the kinematic models controlling the displacement of the manipulator can be divided into two parts: a model for the displacements in the horizontal plane, which is equivalent to a 3-RPR manipulator; -a model for the translations along the vertical axis equivalent to the model for the vertical translations of a pantograph linkage. The type 2 singularities of such a manipulator appear when [30, 31]: a) the three legs of the manipulators are parallel, which is impossible for the developed PAMINSA manipulator. b) the orientation of the platform is equal to ) / ( cos 1 b pl R R -

Fig. 3 .

 3 Fig. 3. Type 2 singular configuration of examined manipulator (horizontal projection of the examined structure).

  platform triangles are respectively equal to R b = 0.35 m and R pl = 0.1 m; magnification factor of the pantograph: k = 3; gravitational acceleration g is equal to 9.81 m/s 2 . lengths of the links of the pantograph linkages: L B1 = 0.308 m, L B2 = 0.442 m, L B3 = L B8 = 0.42 m, L B4 =k L B7 = 0.63 m, L B5 = 0.0275 m, L B10 = 0.3635 m; masses of the joints of the pantograph linkages: m 2 = 0.214 kg, m 3 = 0.338 kg, m 4 = 0.262 kg, m 5 = 0.233 kg, m 7 = 3.08 kg, m 8 = 0.305 kg, m 9 = 0.259 kg; mass of the platform: m pl = 2.301 kg; masses of the links of the pantograph linkages: m B1 =1.221 kg, m B2 = 0.921 kg, m B3 = 0.406 kg, m B4 = 0.672 kg, m B7 = 0.107 kg, m B8 = 0.403 kg, m B10 = 0.436 kg; term of the inertia matrix of the platform:

Fig. 4 .

 4 Fig. 4. Displacement of the PAMINSA along the prescribed straight line divided into two parts.

  (a) actuator M 1 . (b) actuator M 2 . (c) actuator M 3 . (d) actuator M v .

Fig. 5 .

 5 Fig. 5. Input efforts of the PAMINSA in the case of the sixth order polynomial trajectory planning, obtained by the ADAMS software.

Fig. 6 .

 6 Fig. 6. Input efforts of the PAMINSA in the case of the fifth order polynomial trajectory planning obtained by the software ADAMS.

with b 3 =

 3 -3.3033, b 4 = 5.10456, b 5 = -2.45207, b 6 =0.37844, c 1 = 1, c 2 = -13.25829, c 4 = 2365.3672, c 5 = -11953.07236 and c 6 = 16158.76157.

Fig. 7 .

 7 Fig. 7. The built prototype of the PAMINSA with 4 DOF, which was used for the experimental validation of obtained results.

Fig. 8 .

 8 Fig. 8. Trajectory reproduction on the PAMINSA during the displacement of the platform with the fifth order polynomial law (view from below).

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& & & & & & & & & & & & & & & & & & & & & & & &

& & & &


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

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

Victor Glazunov

G C G M -Ea3913

Improvement of functional performance of spatial parallel manipulators using mechanisms of variable structure

Keywords: Parallel manipulator, optimal control, singularity-free zones, pressure angle, force transmission y

A procedure for the increase of singularityfree zones in the workspace of spatial parallel manipulators is presented in this paper. The procedure is based on the control of the pressure angles in the joints of the manipulator. The zones, which cannot be reached by the manipulator, are detected. For increase of the reachable workspace of the manipulator the legs of variable structure are proposed. The design of the optimal structure of the spatial parallel manipulator 3-RPS is illustrated by a numerical simulation.

I. Introduction

It is well-known that the multiple closed chains in parallel manipulators often lead to difficulties in their design and control. One of the most important problems in the design of parallel mechanisms is the study of their singular configurations. The singular analysis has attracted the attention of several researchers and different studies have been published [START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF][START_REF] Zlatanov | Singularity analysis of mechanisms and robots via a motion-space model of the instantaneous kinematics[END_REF][START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF][START_REF] St-Onge | Singularity analysis and representation of the general Gough-Stewart platform[END_REF][START_REF] Pernkopf | Singularity analysis of spatial Stewart-Gough platforms with planar base and platform[END_REF][START_REF] Wen | Singularities in three-legged platform-type parallel mechanisms[END_REF][START_REF] Wolf | Application of line geometry and linear complex approximation to singularity analysis of the 3-DOF CaPaMan parallel manipulator[END_REF][START_REF] Bandyopadhyay | Analysis of configuration space singularities of closed-loop mechanisms and parallel manipulators[END_REF][START_REF] Zhao | Analysis of the singularity of spatial parallel manipulator with terminal constraints[END_REF][START_REF] Arakelian | Singularity analysis of the parallel manipulator PAMINSA[END_REF][START_REF] Glazunov | Classification principles and analysis methods for parallel-structure spatial mechanisms[END_REF]. These studies are often developed for design of parallel mechanisms without singular configurations. It would be very suitable if such a result could be achieved by optimal legs assembly.

However, in the previous works, there is another trend which consists in the elimination of singular zones from the whole workspace of the manipulator by the limitation of the workspace. Thus, the workspace of the parallel manipulators which is less than the serial manipulators becomes smaller and limits their functional performances. This has led some researchers to the problem of the optimal control of the parallel manipulators with singular configurations. Alvan and Slousch [START_REF] Alvan | On the control of the spatial parallel manipulators with several degrees of freedom[END_REF] suggested a solution based on the following considerations: the well known Gough-Steward platform is modified and two legs are added, then the optimal control of the manipulator is carried out by six actuators chosen from eight. As an optimization criterion the algebraic value of the Jacobian matrices and the minimum sum of the rootmean-square value of the input torques are used.

The introduction in the initial system of complementary actuators, which make it possible to eliminate the singular configurations of the parallel manipulator by means of optimal control of the motion, can be exerted. However, it is an expensive solution because of the additional actuators (it is well known that the actuators are one of the most expensive components of manipulators) and the complicated control of the manipulator caused by actuation redundancy.

In our previous work [START_REF] Arakelian | Increase of singularityfree zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF] the optimal control of planar parallel manipulators was studied and a new solution of this problem was proposed. It carried out by using mechanisms of variable structure, i.e. a mechanism whose structure parameters can be altered. With regard to the determination of singularity zones inside the workspace of the manipulator we proposed a kinetostatic approach taking account of the force transmission.

In this paper a similar problem for spatial parallel manipulators is studied and a procedure for the increase of singularity-free zones in the workspace of spatial parallel manipulators is presented.

II. Force transmissivity analysis

Let us consider a spatial parallel manipulator (Fig. 1) with 3 degrees of freedom (two orientations and one translation) [START_REF] Lee | Kinematic analysis of a three DOF in-parallel actuated manipulator[END_REF], which consists of the base A 1 A 2 A 3 , the output link B 1 B 2 B 3 and 3 identical legs composed of one revolute pair A i , one prismatic pair C i and one spherical pair B i (i = 1,2,3). Let us examine the pressure angles of the considered manipulator [START_REF] Balli | Transmission angle in mechanisms[END_REF]. Let the revolute pairs A i be actuated and passive joints be located at B i and C i . Thus, each kinematic chain includes one actuated and two passive pairs. The wrench acting to the output link is reciprocal to the unit vectors situated along the axes of non-actuated pairs. Let E i1 , E i2 , E i3 E i4 , E i5 (Fig. 1) be the unit vectors of the axes of kinematic pairs, where i (i=1,2,3) is the number of the chain. Here E i1 corresponds to rotating actuated pair, E i2 corresponds to sliding passive pair, E i3 E i4 , E i5 correspond to the spherical passive pair. The Plücker co-ordinates of these unit screws can be described in the matrix (E) i .

                                z i y i z i z i y i x i z i y i z i z i y i x i z i y i z i z i y i x i z i y i z i z i y i z i z i
e e e e e e Here E i1 , E i3 E i4 , E i5 are the unit screws of zero pitch, i.e.

0 0 1 1 0 1 1 0 1 1       z i z i y i y i x i x i
e e e e e e , etc., E i2 is the unit screw of infinite pitch,

i Ai Bi x i l x x e / ) ( 0 2   , i Ai Bi y i l y y e / ) ( 0 2   , i Ai Bi z i l z z e / ) ( 0 2  
, x Ai , x Bi , y Ai , y Bi , z Ai , z Bi are the coordinates of the points A i and B i , l i is the distance between the points A i and B i , i=1,2,3. Without interruption of generality we can assume that E i3 is parallel to E i1 , E i4 is parallel to E i2 , and E i5 is perpendicular to E i4 and E i2 .

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

We can obtain the wrenches which are reciprocal to the unit vectors of the axes of the passive kinematic pairs. The conditions of reciprocity are:

  0 1 0 1 0 1 1 1 1 1 z i y i x i x i y i x i i r r r r r r R and   0 2 0 2 0 2 2 2 2 2 z i y i x i x i y i x i i r r r r r r R
(i=1,2,3). They are perpendicular to the axis E i2 and intersect the point B i . Without the loss of generality we can assume that R i1 is perpendicular to E i1 and coincides with E i5 , and R i2 is parallel to E i1 and coincides with E i3 . The coordinates of wrenches in the form of the matrix (R) 6×6 are given by:

                                       z y x z

(R)

In singular configurations the system of the wrenches (R) degenerates.

To find the pressure angles we consider the wrenches R ij and the directions of the velocities of the points B i [START_REF] Glazunov | Classification principles and analysis methods for parallel-structure spatial mechanisms[END_REF]. The velocity of the point B 1 is determined by the equations expressing the Plücker co-

ordinates   z y x z y x v v v 1 1 1 1 1 1    of the
                                       z z y y x x z z y y x x z z y y x x z z y y x x z z y y x x z z x x z z y y x x v r r r v r r r v r r r v r r r v r r r e r e r v r r r              (2)
Here ω 11 is the generalized velocity in the pair A 11 . The left hand sides of the equations ( 2 ...

         
where V B1x , V B1y , V B1z are the co-ordinates of the velocity V B1 of the point B 1 : ,

1 1 1 1 1 1 B z B y x x B y z V V      , 1 1 1 1 1 1 B x B z y y B z x V V      , 1 1 1 1 1 1 B y B x z z B x y V V     
Finally, the pressure angle of leg 1 can be written as: We can find two other pressure angles by a similar way.

              
It was noted that in the singular configurations all the pressure angles are equal to 90°. Indeed, in this case the determinant of the matrix (R) is equal to zero. Therefore the wrench R 11 is reciprocal to the twist Ω , the velocity V B1 is perpendicular to the axis of the wrench R 11 . In this case ω 11 =0.

Thus, the pressure angles can be determined at the joints of each kinematic chain. In this way we could map the maximum value of the pressure angles in the whole workspace of the parallel manipulator to detect the inaccessible zones with unfavourable values of the pressure angles.

If the prescribed path of the parallel manipulator intersects any unacceptable zone in which the pressure angle has an inadmissible value the transmission of the motion can be disrupted. In this case, it is necessary to change the structural parameters of the mechanism, i.e. the input motions.

Fig. 2 shows a schematic of the modified leg with the added articulated dyad which allows changing the input motion. The rotating actuators are mounted on the base and connected by electromagnetic clutches with the links A i D i and A i C i . The input motion can be transmitted either by the link A i D i or A i C i (i=1,2,3). In this way we can obtain the leg of the mechanisms with different structural parameters, which changes the direction of the wrench R i1 and allows increasing the singularity-free zones.

Let us consider the system of wrenches existing in this case. The link B i C i is constrained by two wrenches of zero pitch T i1 and T i2 . The axis of the wrench T i1 is perpendicular to the line A i B i and the axis of the wrench T i2 coincides with the axis of the link C i D i . The unit screw E' i1 of the twist of the link B i C i is reciprocal to the wrenches T i1 and T i2 . This twist is of zero pitch and is parallel to the axis E i1 . The location of the axis E' i1 corresponds to the point of intersection of the wrenches T i1 and T i2 . If the link C i D i is perpendicular to the link B i C i then the wrenches T i1 and T i2 are parallel and the instantaneous motion of the link B i C i is a translation. The wrench R i1 (i=1,2,3) can be determined using the equation analogous to [START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF]. The pressure angle can be found using the equation (3). This approach can be applied for mechanisms with different degrees of freedom and different structures of legs. Particularly at the point A i can be situated a universal joint. Then each kinematic chain determines only one wrench R i whose direction can be changed by choosing different input links. Thus, by such a way, we can determine the pressure angles corresponding to the different structures and obtain all possible workspace with singularity-free zones.

III. Optimal structural architecture of the manipulator taking account of pressure angle

In order to obtain the best structural architecture of the manipulator for a given trajectory, in this section we describe a procedure, which allows determining the optimal system of actuation. For this purpose, at the first time, we would like to show the singularity-free zones in the workspace of the 3-RPS spatial parallel manipulator with modified legs. These zones have been determined by using the maximum acceptable values of the pressure angles.

For numerical simulation we consider a manipulator in which the basis triangle A 1 A 2 A 3 is equilateral with radius 0.35 m (Fig. 1) and the platform B 1 B 2 B 3 also represents an equilateral triangle with radius 0.1 m. For added dyads A i D i =C i D i =0.25 m, the articulated dyads are always located on the top of the prismatic pairs as it is shown in Fig. 2 and the translations of the prismatic pairs are limited relative to the joints A i and

C i by values (A i C i ) min =(B i C i ) min =0
.05 m. The origin of the fixed base frame (Oxyz) is located at the centre of the equilateral triangle A 1 A 2 A 3 , the vertical axis z is orthogonal to this triangle and the screws E i1 are tangent to the circle passing through

A 1 A 2 A 3 .
This manipulator has three degrees of freedom and only three of the six position/orientation variables are independent. In this work, as the output parameters are defined two orientation angles  1 ,  2 and the vertical position z of the platform. The angles  1 ,  2 can be obtained by expressing the directional cosines in terms of x-y-x Euler angles  1 ,  2 ,  3 (see the basic kinematic equations in [START_REF] Lee | Kinematic analysis of a three DOF in-parallel actuated manipulator[END_REF]).

Taking into account that the modified manipulator can be actuated either by links A i D i or by links A i C i , for given output parameters (z, 1 , 2 ) of the platform, we have 8 different combinations of actuation, i.e. we have 8 different combinations of input parameters presented below (underlined letters show the input pairs, "R" for input links A i C i with input angles  i and "P" for input links A i D i with input displacements  i ):

RRR: RPS-RPS-RPS: q (1) =( 1 , 2 , 3 ) RRP: RPS-RPS-RPS: q (2) =( 1 , 2 , 3 ) RPR: RPS-RPS-RPS: q (3) =( 1 , 2 , 3 ) RPP: RPS-RPS-RPS: q (4) =( 1 , 2 , 3 ) PRR: RPS-RPS-RPS: q (5) =( 1 , 2 , 3 ) PRP: RPS-RPS-RPS: q (6) =( 1 , 2 , 3 ) PPR: RPS-RPS-RPS: q (7) =( 1 , 2 , 3 ) PPP: RPS-RPS-RPS: q (8) =( 1 , 2 , 3 ) Table 1 shows the workspaces of each case of actuation for the altitude of the platform equal to 0.1 m. In these figures, several zones can be seen, which correspond to the variations of the maximum values of the pressure angle for given orientation ( 1 , 2 ) of the platform. The contrast intensity shows the variations of the pressure angle (see Fig. 3). Fig. 4 shows the reachable workspace of the modified parallel mechanism with legs of variable structure. We can see that the workspace of the modified manipulator is only composed of singularityfree zones and the whole workspace of the manipulator is reachable (increase until 100%). Now we would like to describe a procedure, which allows determining the optimal system of actuation. This algorithm is based on the control of the pressure angles in the joints of the manipulator along the given trajectory. It is similar to the procedure given in [START_REF] Arakelian | Increase of singularityfree zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF] for planar parallel manipulators.

At first the calculation of the pressure angles in the joints along the trajectory for all possible structures of the parallel mechanism with variable architecture must be accomplished, then the best structure must be chosen for which the maximum value of the pressure angle along the trajectory is always less than the limit value. If there is no structure satisfying this condition, the given trajectory must be decomposed in several parts and the generation of the motion must be carried out by different structures. It is obvious that in this case it would be desirable that the trajectory can be realized by minimal structural changes.

A numerical example is considered below in order to illustrate the application of the suggested design procedure.

For given parallel manipulator (Fig. 1) with legs of variable structure (Fig. 2) generate the trajectory from the initial position P 1 (z=0.3 m,  1 =0 rad.,  2 =0 rad.) to the final position P 2 (z=0.3 m,  1 =0 rad.,  2 =1 rad.), keeping z and  1 constant.

Estimation of the pressure angle along the given trajectory shows that the best structural solution for generation of motion is the RPS-RPS-RPS mechanism, i.e. when the first actuator is connected with the link A 1 D 1 and two others with the links A 2 C 2 and A 3 C 3 . In this case the maximum values of the pressure angles in the joints are always less than the limit value.

In order to illustrate the variations of torques for examined case we develop a model of the manipulator with the given trajectory using the ADAMS software.

A force parallel to the z-axis and equal to 100 N was applied to the platform and the friction coefficients in the prismatic pairs were equal to 0.01. The obtained torques are shown in Fig. 5. We can note that the torques have admissible values along the trajectory.

It is obvious that the similar mechanisms of variable structure can also be designed on the base of the screw or cam systems, the rhombic pantographs, etc.

IV. Conclusion

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

Fig. 1 .

 1 Fig. 1. Spatial parallel manipulator 3-RPS.

  coordinates of the wrenches to be found. Equations[START_REF] Gosselin | Singularity analysis of closedloop kinematic chains[END_REF] mean that each connecting kinematic chain determines two

  ) express the reciprocal moments of the twist Ω 1 and the wrenches R 11 , R 12 , R 21 , R 22 , R 31 , R 32 . According to the last five equations the twist Ω 1 is reciprocal with the wrenches R 12 , R 21 , R 22 , R 31 , R 32 . The right hand sides of the first and second equations correspond to reciprocal moments of the twist E 11 ω 11 and the wrenches R 11 and R 12 . As the wrench R 11 is of zero pitch and parallel to E 11 then the reciprocal moment of this wrench and the twist E 11 ω 11 can be written as: the wrench R 12 is perpendicular to the E 11 then the reciprocal moment of this wrench and the twist E 11 ω 11 is equal to zero: wrenches R 11 , R 12 , R 21 , R 22 , R 31 , R 32 are of zero pitch therefore the reciprocal moment of the twist Ω 1 and the wrench R 11 can be written as:

Fig. 2 .

 2 Fig. 2. Planar representation of the leg with variable structure.

Fig. 3 .

 3 Fig. 3. The contrast intensity corresponding to the pressure angle values.

Fig. 4 .

 4 Fig. 4. The reachable workspace of the spatial parallel manipulator with modified legs (z = 0.1 m).

Fig. 5 .

 5 Fig. 5. Torques of the actuators.

twist Ω 1 existing

  by fixed actuated pairs A 2 and A 3 .

	11	1	11	1	...	11	1	11		11	11	...	11	0 11	
					...		0
	12	1	12	1		12	1
					...		0
	21	1	21	1		21	1
					...		0
	22	1	22	1		22	1
					...		0
	31	1	31	1		31	1
					...		0
	32	1	32	1		32	1

12th IFToMM World Congress, Besançon (France), June 18-21, 2007


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

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

G C G M -Ea3913

Singularity Analysis of PAMINSA Manipulators

Keywords: Parallel manipulator, decoupled motions, kinematics, singularity analysis

PAMINSA (PArallel Manipulator of the I.N.S.A.) is a new family of parallel manipulators from four to six degrees of freedom (DOF), which have been developed at the I.N.S.A. in Rennes. The particularity of these manipulators is the decoupling of displacements in the horizontal plane from its translation along the vertical axis. Such a decoupling improves some mechanical properties of the manipulator making it more efficient. In this paper a singularity analysis of PAMINSA with four, five and six degrees of freedom is presented. The nature of each kind of singularity is discussed.

I. Introduction

The parallel manipulators have been under intensive study for over a decade and different approaches and solutions have been developed and documented. In the recent years, the decoupling of motions of parallel manipulators is motivated by continuous interest researches in the solution of problems for control simplification. It is obvious that the complex nonlinear dynamics of the traditional parallel manipulators complicates their control and one mechanism with a linear input/output relation is more appealing for industrial applications. For this purpose, the different parallel structures [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] have been developed in which all input/output equations are linear and, consequently, all motions are fully decoupled. However, 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.

Another trend of the kinematic decoupling is proposed in the design of PAMINSA [START_REF] V. Arakelian | Parallel robot comprising means for setting in motion a mobile element split in two separate subassemblies[END_REF], which consists in decoupling the motions in the horizontal plane and the translation along the vertical axis. In other words, in the design of the PAMINSA, a partial decoupling of motions is achieved, which provides several mechanical advantages. Mechanical, kinematic and dynamic characteristics, as well as the mechanical properties of the invented manipulator, have been studied in our previous works [START_REF] Arakelian | A New Decoupled Parallel Manipulator with Four Degrees of Freedom[END_REF][START_REF] Briot | Le PAMINSA : un nouveau manipulateur d'architecture parallèle aux mouvements découplés[END_REF][START_REF] Arakelian | Static and dynamic analysis of the PAMINSA[END_REF][START_REF] Arakelian | Design and prototyping of new 4-, 5-and 6-DOF parallel manipulators based on the copying properties of the pantograph linkage[END_REF] and a prototype of PAMINSA with 4-DOF (Fig. 1) was built in the I.N.S.A. We would like to renote in few words the obvious advantages of the suggested manipulator architecture: (i) 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; (ii) the cancellation of static loads on the rotating actuators which displace the platform in the horizontal plane; (iii) a great accuracy in the horizontal positioning because the payload can be locked in the horizontal plane by mechanical architecture of the manipulator; (iv) the use of the less powerful actuators for the generation of motions on the horizontal plane; (v) 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).

In this paper the singularity analysis of the different types of PAMINSA from 4 to 6 degrees of freedom is presented and kinematic interpretations of the singular configurations are provided. The organization of this paper is as following. The second section describes the kinematics of the PAMINSA manipulators with different degrees of freedom. The third section uses the obtained Jacobian matrices for the singularity analysis of the manipulators and kinematic interpretations of the singular configurations are given.

II. Kinematics of the PAMINSA manipulators

The family of the PAMINSA manipulators [START_REF] Arakelian | Design and prototyping of new 4-, 5-and 6-DOF parallel manipulators based on the copying properties of the pantograph linkage[END_REF] is based on the copying properties of the pantograph linkage, which composes the kinematic chain between the fixed base of the manipulator and its end-effector (Fig. 2). 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. The PAMINSA-6D3L (Fig. 3a) is a manipulator with 6 degrees of freedom composed of 3 legs. 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 rotations of the three legs allow the displacements of the platform on the horizontal plane (2 translations and one rotation about the vertical axis). The point B i of each leg is controlled by an independent actuator. The vertical translation is controlled by the simultaneous vertical displacements of points B i of the 3 legs. The orientations about two other axes is obtained by the vertical displacements of the point B i of the legs.

For the PAMINSA-5D3L (5-DOF -Fig. 3b), 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 translation and one inclination of the platform are obtained as previously. 12th IFToMM World Congress, Besançon (France), [START_REF]12th IFToMM World Congress[END_REF] For the PAMINSA-4D3L (4-DOF -Fig. 3c), the rotations of the three legs allow the displacements of the platform in the horizontal plane. 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.

It should be noted that the PAMINSA with 5-DOF can be assimilated to a PAMINSA with 6-DOF whose first two linear actuators have the same displacements. Thus one inclination of the platform is prevented. In a similar way, the PAMINSA with 4-DOF is a PAMINSA with 6-DOF whose whole linear actuators have the same displacements. Whole inclinations of the platform are prevented and only one rotation about the vertical axis is allowed.

Thus, in the general case, the kinematics of the PAMINSA-6D3L describes the kinematics of the other manipulators. The position of the centre of the platform P and the orientation of the moving frame {M} in the base frame {B} (x-axis is collinear to O 1 O 2 and z-axis is vertical) are represented by x=(x,y,z,φ,ψ,θ) T

III. Singularity analysis

The singularity analysis has attracted the attention of several researchers and different studies have been published [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Zlatanov | Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics[END_REF][START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF][START_REF] St-Onge | Singularity analysis and representation of the general Gough-Stewart platform[END_REF][START_REF] Pernkopf | Singularity analysis of spatial Stewart-Gough platforms with planar base and platform[END_REF][START_REF] Wen | Singularities in three-legged platform-type parallel mechanisms[END_REF][START_REF] Wolf | Application of line geometry and linear complex approximation to singularity analysis of the 3-DOF CaPaMan parallel manipulator[END_REF][START_REF] Bandyopadhyay | Analysis of configuration space singularities of closed-loop mechanisms and parallel manipulators[END_REF][START_REF] Zhao | Analysis of the singularity of spatial parallel manipulator with terminal constraints[END_REF][START_REF] Glazunov | Classification principles and analysis methods for parallelstructure spatial mechanisms[END_REF]. Zlatanov in [START_REF] Zlatanov | Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics[END_REF] presented a method that can identify the singularities of both passive and active chains by studying the deficiency of the rank of an augmented non-square Jacobian matrix. However this analysis is quite difficult and not useful for the PAMINSA manipulators. The singularity analysis presented here is carried out by Gosselin and Angeles approach [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF], based on the properties of the Jacobian kinematic matrices of the mechanical structure, i.e. when the Jacobian matrices relating the input velocities and the output velocities become rank deficient.

PAMINSA are parallel manipulators in which singular configurations can be separated into two cases: singularities of the pantograph linkage used as a leg and singularities of the simplified schematic representation of the PAMINSA manipulators in which the pantograph mechanism is replaced by a chain PRPS (Fig. 4). The pair S' i corresponds to the free translational displacement of both prismatic pair S i and pantograph linkage. The actuators M' vj correspond to actuators M vj whose displacement is amplified by the magnification factor of the pantograph. In PAMINSA manipulators, these singularities are not coupled and can be examined separately.

It may be noted that the singular configurations of pantograph linkage can be found by an analysis of the articulated parallelogram. They are well known and we shall not deal with. The study below is only devoted to the singularities of the 3-PRPS parallel structure. PAMINSA-6D3L: Differentiating equations (1-6) with respect to time, we obtain a 6 dimensional system:

0 Bt q A = + & (7) (j=i) ⎩ ⎨ ⎧ = - = - - - = 0 0 ) ( ) ( Ci i i Oi Ci i Oi Ci j z Z k c y y s x x f β β (j=i+3)
12th IFToMM World Congress, Besançon (France), [START_REF]12th IFToMM World Congress[END_REF] where

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ = k sym k k q f i j 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 1 ρ ρ ρ A ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - - - - - = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ =
θ ψ φ t T Z Z Z ] , , , , , [ 3 2 1 3 2 1 β β β = q 2 2 ) ( ) ( Oi Ci Oi Ci i y y x x - + - = ρ ( i=1,2,3) ) ( ) ( ) ( ) ( ) ( 4 i i i i n i s s c c c R d β φ γ θ ψ β φ γ θ + + + + + = (i=1,2 ,3)
Singularity of parallel manipulators appears when matrices A and B are rank-deficient. We will deal only with the singularities of the first and the second type. Third type singularities are a mix a both Type 1 and 2 singular configurations.

Examining matrix A and B of PAMINSA with 6-DOF, it appears that:

3 2 1 3 ) det( ρ ρ ρ k = A ( 8
) ) 8 /( 27 ) det( 3 2 1 3 ρ ρ ρ ψ ψ ∆ = s c R n B ( 9
)
where ∆ is the expression of a conic given in appendix A.

First of all, it should be noted that the expressions of det(A) and det(B) do not depend of the altitude z of the platform. det(A) = 0 involves that the manipulator is in Type 1 singularity. This expression is achieved when ρ i is equal to 0 (i=1,2,3), i.e. points O i , B i and C i are aligned. In such configuration, one rotation of the input link M i cannot bring to the displacement of the platform (Fig. 5).

The study of det(B) is much more complicated. This expression is the product of many terms which have to be conscientiously analyzed. [START_REF] Arakelian | Design and prototyping of new 4-, 5-and 6-DOF parallel manipulators based on the copying properties of the pantograph linkage[END_REF] (b) the inclination angle ψ is equal to ±π/2; in such configuration, the rotation about the axis x of angle ψ is impossible and small rotations of the platform are allowed (Fig. 6a). (c) for any fixed altitude, the platform is on a conic ∆ = 0 located in the horizontal plane. The coefficients of ∆ depend on the orientation angles of the platform. Thus, for given values of angles φ, ψ and θ, this singularity is only due to a particular configuration of actuators M 1 , M 2 and M 3 . It appears when the three wrenches R i1 [START_REF] Dimentberg | The screw calculus and its applications in mechanics[END_REF] (applied on the platform by the actuators M i and located on the horizontal plane-see Fig. 4) intersect a unique line W perpendicular to the horizontal plane (Fig. 6b). In such case, this line is the instantaneous axis of rotation of the moving plate;

(d) the length ρ 1 , ρ 2 or ρ 3 tend to +∞ , what means that the legs of the manipulator are parallel. Thus the platform is able to translate along the direction of the passive prismatic pairs S' i (Fig. 6c).

PAMINSA-5D3L:

As previously noted, the PAMINSA with 5-DOF can be assimilated to a PAMINSA with 6-DOF whose first two linear actuators have the same displacements. Thus angle θ is equal to 0. To find its singularity loci, these constraints have to be introduced in the expressions of det(A) and D.

(i=1,2,3) The kinematic interpretation of these singularities is the same as for the previous manipulator. Finally, we would like to note that, in the present work, the singularity analysis was carried out only by taking into account the kinematic relationships. In practice, this problem is much more complicated and it may be studied with kinetostatic and dynamic aspects. Among several effects, which can modify the singularity analysis conditions, we would like to note the friction. The singularity analysis of parallel manipulators, taking into account the friction in the joints, has been developed in our previous work [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF].

⎩ ⎨ ⎧ - = + + + ) ( ) ( ) ( 5 i n i i n i s c R s c s R d γ θ ψ γ θ β φ ψ (i=4,5,6) (i=1,2,3) ⎩ ⎨ ⎧ - + - = + + + + + ) ( ) ( ) ( ) ( ) ( 6 ) ( i n i i i i n i c s R c c c s s R d γ θ ψ β φ γ θ ψ β φ γ θ (i=4,
φ cos 2 2 2 2 2 b n n b R R R R y x - + = + (

IV. Conclusions

In this paper the singularity analysis of PAMINSA with four, five and six degrees of freedom has been presented. The singularities have been determined in analytic form by algebraic approach based on the analysis of the properties of the Jacobian matrices. The nature of each kind of singularity has been discussed and kinematically analyzed. In practice, the obtained results can be very useful in the selection of the optimal architectures of PAMINSA and in the problems of its optimal control. θ

ψ θ φ ψ ψ + - - + - + - + + - - + - + - + - - = s R c c s c s c c s s c c c s c R R c c s c s c c c s c c R c D b b n n ) ) 1 2 ) 1 ( 2 ) 2 ) 4 2 2 ( ( ( )) 4 3 ( )) 4 3 ( ) 4 1 )( 1 ( ( ( ( ) 1 (

Fig. 1 .

 1 Fig. 1. The prototype of the PAMINSA with 4-DOF.

Fig. 2 .

 2 Fig. 2. Schematics of one leg of PAMINSA manipulators For each kind of PAMINSA, 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 . Displacements of point C i are amplified by the magnification factor of the pantograph.The PAMINSA-6D3L (Fig.3a) is a manipulator with 6 degrees of freedom composed of 3 legs. The points A i are actuated on the horizontal plane by 3 motors M i which control directly the rotation of the

Fig. 3 .

 3 Fig. 3. Schematics of PAMINSA manipulators.

  and the actuated variables by q=(β,,β 2 ,β 3 ,Z 1 ,Z 2 ,Z 3 ) T . Parameters x, y, z, φ, ψ , θ represent respectively the three components of the position of point P and the three rotation angles of the platform. The angles φ, ψ and θ can be obtained by expressing the directional cosines in terms of z-x-z Euler angles φ, ψ, θ. Parameters β 1 , β 2 , β 3 , Z 1 , Z 2 , Z 3 represent respectively the rotations of the three legs of the manipulator about the z-axis of frame {B} and the vertical position of points B i . Note that for the following analyses, O 1 O 2 O 3 and C 1 C 2 C 3 represent equilateral triangles. The closed loop relations relative to x and q can be expressed as: for (i=1,2,3) (1-6) where -c α and s α represent the cosines and sinus of angle α respectively (α = β i , γ i , φ, ψ and θ) -k is the magnification factor of the pantograph -R n and R b are the platform and base radii respectively -i=1,2,3) with Rot(α,w) the matrix representing the rotation of angle α around the w-axis of the intermediate frame (w = x, y and z).

Fig. 4 .

 4 Fig. 4. Simplified schematic representation of the i-th actuated leg.

Fig. 5 .

 5 Fig. 5. Example of first type singularity. Thus, det(B) = 0 if: (a) the inclination angle ψ is equal to 0 (or π); such case is physically meaningless. This is a formulation singularity due to the use of the Eulerangles formulation [21]. If ψ = 0 (or π), rotations of angles φ and θ are equivalent (about the same zaxis). Thus Type 2 singularities only correspond to cases (b), (c) and (d) and appear when: 0 ) 8 /( 27 3 2 1

  Fig. 6. Examples of second type singularities As for the previous case, the manipulator is in Type 1 singularity when ρ i is equal to 0 (i=1,2,3), i.e. points O i , B i and C i are aligned. Furthermore, the manipulator is in Type 2 singularity when: (a) the inclination angle ψ is equal to ±π/2. This case correspond to figure 6a; (b) for any fixed altitude, the platform is on a conic Λ = 0 located in the horizontal plane. The coefficients of Λ depend on the orientation angles φ and ψ of the platform. Their expressions are given at appendix B. This case corresponds to figure 6b; (c) the length ρ 1 , ρ 2 or ρ 3 tend to ∞ . This case correspond to figure 6c.

  PAMINSA-4D3L: Similarly to the previous case, the PAMINSA with 4-DOF can be assimilated to a PAMINSA with 6-DOF whose linear actuators have the same displacements. Thus angles ψ and θ are equal to 0. These new constraints have to be introduced in the expressions of det(A) and D.The Type 1 singularities are not different from the previous manipulators. The Type 2 singularities appear when:(a) the angle φ is equal to for any fixed altitude, the platform is situated on a circle located in the horizontal plane, whose radius depends on angle φ. The expression of this circle is:

  11) (c) the length ρ 1 , ρ 2 or ρ 3 tend to +∞ . Cases (a) and (b) correspond to figure 6b and case (c) to figure 6c.

12th IFToMM World Congress, Besançon (France),[START_REF]12th IFToMM World Congress[END_REF]

& & & & & & =

& & & & &


Simon Lessard

Pascal Bigras

Ilian A Bonev

Sébastien Briot

Vigen Arakelian

Vigen Arakelian Optimum

Static Bal- Ancing

L G C G M -Ea3913

Optimum Static Balancing of a Parallel Robot for Medical 3D-Ultrasound Imaging

Keywords: static equilibrium, parallel robot, medical robot, root-mean-square minimization

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

Recently, parallel robotics has broken through new field of research such as high speed manipulation, material handling, motion platforms and medical equipment. In the latter, a new robot has been developed at École de technologie supérieure (Fig. 1). This robot is designed to perform an ultrasound scan of a human patient's arteries. It consists of two five-bar planar mechanisms, which are connected to a tool holder by an articulated telescopic strut. Four motors are mounted inside the main frame while a fifth motor is attached directly to the tool holder allowing large amplitude tool rotations. The whole frame is mounted on a linear motor which allows large horizontal displacements of the robot. Thus, such an architecture has six degrees of freedom and ensures the required geometric and kinematic characteristics (see tables I and II) [START_REF] Lessard | Parallel Robot for Medical 3D-Ultrasound Imaging[END_REF].

The robot workspace was defined considering the examination of one half of the human body, cut vertically * E-mail: simon.lessard.3@ens.etsmtl.ca A partnership with Institut National des Sciences Appliquées (Rennes, France) has led to this present work on optimum static balancing of the six-degree-of-freedom medical robot.

(Fig. 2). A robot suited for the quantification of the lower limbs arteries could perform any less restricting arterial examination since it is the longest arterial examination. The linear displacement was chosen long enough to cover the whole body from neck down, thus preventing repositioning of the patient for different examinations or having to place him meticulously in the robot workspace. Designing a robot for a precise confined workspace would have led to a less practical system. The first virtual prototype version of the robot (Fig. 1) is smaller resulting in a smaller workspace. For assembly considerations, symmetric motors of each five-bar mechanism are not mounted vis-à-vis.

The robot design next step is the minimization of the input torques. This goal is well-known in the field of robotics and is composed of the static balancing of moving masses.

Static balancing is typically achieved by adding a counterweight or a spring to each bar. Complete static balancing is carried out by keeping constant the potential energy of the system for all configurations. In previous studies, a number of methods have been proposed for gravity balancing of robotic systems: spring balancing of a planar pantograph [START_REF] Herder | Energy-Free Systems. Theory, conception and design of statically balanced mechanisms[END_REF][START_REF] Shin | Spring equilibrator theory for static balancing of planar pantograph linkages[END_REF], n-springs solutions to a single attached point [START_REF] Walsh | Spatial spring equilibrator theory[END_REF], balancing of a spatial positioning table [START_REF] Wang | Static balancing of spatial threedegree-of-freedom parallel mechanisms[END_REF] and 6-dof parallel manipulators [START_REF] Ebert-Uphoff | Static balancing of spatial parallel platform mechanisms -revisited[END_REF][START_REF] Gosselin | Static balancing of spatial six-degreeof-freedom parallel mechanisms with revolute actuators[END_REF], as well as several auxiliary mechanisms for spring support [START_REF] Fattah | On the design of a passive orthosis to gravity balance human legs[END_REF][START_REF] Fattah | Gravity-Balancing of Classes of Industrial Robots[END_REF][START_REF] Simionescu | Static balancing of the industrial robot arms. Part I: discrete balancing[END_REF][START_REF] Simionescu | Static balancing of the industrial robot arms. Part II: continuous balancing[END_REF][START_REF] Wang | Passive mechanisms with multiple equilibrium configurations[END_REF].

This article deals with the optimum static balancing of a parallel robot developed for medical 3D-ultrasound imaging. The suggested analytical solution allows a significant reduction in the input torques by means of simpler design solutions. The paper is organized as follows. Firstly, the input torques due to the static and dynamic loads are examined and a simplified calculation approach is proposed. Then an optimal static balancing method is developed, which is formulated by the input torques root-mean-square values minimization.

II. Static and Dynamic Models

In our medical robotic application, motion is slow and smooth. Therefore, the end-effector velocity and acceleration are small. Our observations show that the input torques caused by the inertia effect of the moving masses are very small compared to the input torques caused by the gravity force. Thus, inertia effect can be neglected. For example, the variation of the input torques for a prescribed trajectory (with 0.6m/s 2 maximal acceleration) shows that the maximal difference between the dynamic and the static loads is less than 1%. This means that the motor torques are mainly caused by gravity.

III. Complete Static Balance

Counterweights mounted on each moving links achieve complete static equilibrium. Fig. 3 illustrates the complete static balance of the parallel robot for medical 3Dultrasound.

Fig. 3. Complete balancing by counterweights

Clearly, the added counterweights increase the robot's inertia. Also, the tool holder motor would be difficult to assemble with a counterweight attached because the joint is not a ball joint, but a combination of a universal joint and a revolute joint.

In the case of complete balancing with springs, it is necessary to add auxiliary mechanisms (parallelograms), which lead to a very complicated mechanical architecture. A mechanical system that is completely balanced becomes either very heavy or too complicated to manufacture and assemble. That is why we developed a partial balancing approach. It is obvious that the load reduction on actuators is partial, but the design solution is very simple.

IV. Input Torques

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

( ) ) 6 ,..., 1 ( 13 1 = ⋅ = ∑ = i j j i T j i G J τ (1)
where (J T j ) i corresponds to the i th line of the transposed Jacobian matrix between the center of masses S j of the link j and the actuated variables q i . G j is the gravity forces of the link j (Fig. 456).

The structure of this robot has some particularities which allow a considerable simplification of torques determination. The tool holder represents an Assur group [START_REF] Zinoviev | Théorie des mécanismes et des machines[END_REF] with two links and three joints (spherical, universal and revolute pairs) dividing the robot architecture in two parts. Thus the tool holder can be disconnected from the robot structure to be examined separately. From this point forward, the tool is considered statically balanced around the sixth motor, i.e. the tool's center of mass coincides with the motor rotation axis. Such an approach simplifies reaction forces and input torques calculations.

The reaction forces R E and R H (respectively applied by the tool holder at points E and H) are function of the yaw and pitch angles only, the roll angle being equal to the sixth motor rotation angle, this motor input torque being independent of the other motors. The length L of the passive linear joint is calculated as a function of the orientation angles:

α β cos cos f L L = (2)
where f L is the distance between the two planar five bar mechanisms:

H E f x x L - = .
Thus, the vertical reaction forces on point H and E can be determined from the static equilibrium equations:

L g r m r m r L m R Hz ) ) ( (
+ + - = (3) 6 1 1 1 2 1 3 ( ) Ez H z R m m m m g R = + + + - (4)
where m i , and r i are respectively the mass and position of the centre of masses of the i th link, g is the gravity and m 6 is the mass of actuator 6.

The horizontal reaction forces are projected from axis w to axis y: Thus, it is possible to deduce the potential energy V of the first five bar mechanism (Fig. 6):

α β sin tan ) ( 12
Hz Hy R g m R - = ( 5
)
Hy Ey R R - = (6)
E Ey E Ez y R z R g z m z m z m z m V + + + + + = ) ( 8 8 7 7 3 3 2 2 (7) with 2 2 2 cos q r z z A + = ( 8
) 3 3 3 cos q r z z C + = (9) E E A z L r z z q L z + - + = 7 7 2 2 7 ) cos ( ( 10
) E E C z L r z z q L z + - + = 8 8 3 3 8 ) cos ( (11)
where z i is the vertical coordinate of the centre of mass S i of the i th link and r 2 =AS 2 , r 3 =CS 3 , r 7 =ES 7 , r 8 = ES 8 .

Then the input torques are determined by differentiating the potential energy by the motor articulation position vector Q. To transform the reaction forces at the tip into reaction motor torques, a Jacobian matrix is needed [START_REF] Clavel | The Lagrange-Based Model of Delta-4 Robot Dynamics[END_REF]:

E J Q τ ∂ ∂ - ∂ ∂ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = V V T 3 2 τ τ (12)
where τ i is the i th actuator torque, [ ]

3 2 , q q = Q
and J is the five-bar mechanism's Jacobian matrix [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. The obtained expression of τ is given in appendix 1.

The second five-bar mechanism's torques τ 4 and τ 5 of actuators 4 and 5 are determined in a similar way.

V. Input Torques Root-Mean-Square Values Minimization

The input torques minimization is carried out by each actuator torque root-mean-square values minimization for all the workspace. The workspace used in the calculation is only the yz plane because the linear motorized axis is not subject to gravity balancing. Three solutions are considered for optimum balancing.

A. Tension / Compression Spring Equilibrium

Firstly, one solution is an equilibrium mechanical system composed of zero free length springs (tension or compression) attached on each motorized arms. The fixed end of each spring is positioned optimally to release the actuator. The spring applied torque's potential energy is:

2 i ki ki ki ki i 2 ki ki ki i K V [ (z c o sq l c o sq) 2 (z sin q l sin q ) ] = - + - ( 13
)
where K i is the stiffness spring coefficient, which is attached on a fixed arm of length z ki and orientation q ki (Fig. 7) and q i is the angular position of the i th actuator.

The other end is linked on the motorized arm at length l ki . By differentiating equation ( 13), we determine the input torque:

) sin( ki i ki ki i i ki ki q q z l K q V - = ∂ ∂ = τ (14)
The spring stiffness coefficient K i and attached linear positions z ki and l ki are consolidated into parameter C i . Thus, the optimal calculated C i will position precisely a certain spring K i on the arms:

) sin( ki i i ki q q C - = τ (15)
Two parameters have to be optimized for each motor: the constant C i and the angular position of the fixed arm q ki .

The torque root-mean-square value minimization leads to the following condition:

) ( ∑ + = WS ki i i f τ τ ki i q C , min → (i=2,…,5) (16)
where WS := Workspace. For this purpose, these conditions must be satisfied:

0 = ∂ ∂ i i C f and 0 = ∂ ∂ ki i q f (17)
from which the coefficients are solved:

) arctan( 2

mi ki t q = (m=1,…,6) (18) ∑ ∑ - - - = WS ki i WS ki i i i q q q q C ) ( sin ) sin( 2 τ (19)
where t mi are the roots of a polynomial p i (t) given in appendix 2.

B. Torsion Spring Equilibrium

Secondly, static balance can be achieved using torsion spring mounted on each actuator axes. The spring torque is linear to the bending angle:

) ( i ki ki q q K - = τ (20)
∑ + = WS ki i i f τ τ ki i q K , min → ( i=2,…,5) (21) 0 = ∂ ∂ i i K f and 0 = ∂ ∂ ki i q f (22)
from which the coefficients are solved:

∑ ∑ ∑ ∑ ∑ ∑ ∑ - - = WS i WS i WS i i WS i WS i WS i WS i i ki q q N q q q q τ τ τ τ 2 (23) ∑ ∑ - - - = WS i ki WS i ki i i q q q q K 2 ) ( ) ( τ (24)
where N is the number of calculated positions in the workspace.

C. Counterweight Equilibrium

Lastly, a balancing approach is carried out by adding a counterweight of mass M i on each motorized axis. The mass M i is placed at a certain angle so that the maximum torque applied is at a specific desired motor angle. The optimum counterweight system is then function of the mass M i and position q ci for a given length

l ci . ) sin( ci i ci i ci q q l M - - = τ (25)
The mass M i and length l ci are consolidated into a single constant C i :

) sin( ci i i ci q q C - - = τ (26)
Determination of parameters C i and q ci is equivalent to the determination of parameters C i and q ki in section A.

VI. Results

The proposed robot (Fig. 1) with above mentioned geometrical parameters and mass distribution (appendix 3) was used for numerical simulation. Three mechanical solutions were tested. As a mean of comparison, reductions of the RMS and maximum motor torques are given in tables (III)-(V). The static torque root-mean-square sum minimization was reduced up to 50% ÷ 91.5%. In more practical terms, the maximum motor torque required was reduced up to % 7 . 46 % 3 . 34 ÷ . Fig. 10 shows input torques variations for unbalanced and optimum balanced robots with extension springs (the simulation was carried out in a static mode or operation).

VII. Conclusion

The perfect static balancing of a spatial multibody mechanical system can eliminate completely the load caused by the gravity force. However, such a solution leads to inevitably complicated design add-ons or to unavoidable increase in total mass. In most cases, the complete balance is achieved only in theory but not in practice. In this paper, a parallel robot for medical 3Dultrasound imaging was optimally statically balanced. The proposed simple solutions presented a partial balancing effect minimizing the actuator torques.

Future work will investigate more efficient minimization by increasing the number of variables to the problem. The development of a prototype with the suggested balancing system is planned.

3 2 3 2 2 2 2 3 3 3 C A E C E E A E E z z y q q L L q z z q y L q z z q y L A - + - ⋅ + - + - - - + - = ) / ( 7 7 2 7 2 2 1 L r L m r m g M + = ) / ( 8 8 3 8 3 3 2 L r L m r m g M + = ) / ) ( /

Fig. 1 .

 1 Fig. 1. CAD model of the prototype parallel robot

12thFig. 4 .

 4 Fig. 4. Euler angles of the tool

Fig. 5 .

 5 Fig. 5. Schematic of the tool assembly

Fig. 6 .

 6 Fig. 6. Schematics of the two five-bars mechanisms with applied reaction forces

  Congress, Besançon (France),June18-21, 2007

Fig. 7 .

 7 Fig. 7. Spring balancing

2

 2

Fig. 8 .

 8 Fig. 8. Torsion spring balancing The minimization conditions are the same: 2 ) (

12th

  Fig. 9. Counterweight balancing

  = 1.7 o C 2 = 0.104 Nm 57.6% 34.7% 3 q k3 = -42.7 o C 3 = 0.263 Nm 91.2% 45.6% 4 q k4 = 3.4 o C 4 = 0.081 Nm 50.1% 33.1% 5 q k5 = -43.0 o C 5 = 0.187 Nm 90.2% 43.1%

5 Fig. 10 .

 510 Fig. 10. Variation of the actuator torques for the orientation angles α=β=γ=0 deg

9 .

 9 81 m/s², m 2 = m 4 = 1.235kg, m 3 = m 5 = 1.549kg, m 6 = 0.331kg, m 7 = m 8 = m 9 = m 10 = 0.536kg, m 11 = 0.107kg, m 12 = 0.083kg, m 13 = 0.111 kg, L 2 = L 4 = 0.5m, L 3 = L 5 = 0.7m, L 7 = L 8 = L 9 = L 10 = 0.6m, L f =0.1m, z A = z F = 0.13m, z C =z K = 0.28m, r 2 =r 4 = 0.2094m, r 3 = r 5 = 0.3046m, r 7 = r 8 = r 9 = r 10 = 0.3m, r 11 = 0.0355m and r 12 = 0.0315m.

TABLE I .

 I Positioning workspace dimensions

	Axis	Orientation (degree)	Description
	Around X	-75° to 90°	Span (arteries axis)
	Around Y	-50° to 35°	Lateral
	Around Z	-45° to 45°	Vertical

TABLE II .

 II Workspace orientations 12th IFToMM World Congress, Besançon (France), June18-21, 2007 Fig. 2. Required workspace for the robotized ultrasound examination

TABLE III .

 III Tension / compression spring optimum configurations K 2 = -4.55 Nm/rad 56.9% 45.4% 3 q k3 = 180 o K 3 = 5.29 Nm/rad 90.4% 45.2% 4 q k4 = 6.2 o K 4 = -3.33 Nm/rad 50.0% 46.7% 5 q k5 = 180 o K 5 = 3.74 Nm/rad 89.2% 34.3%

	Motor	Torsion spring Parameters	RMS	Max Torque
	2	q k2 = 7.7 o

TABLE IV .

 IV Torsion spring optimum configurations

	Motor	Counterweight Parameters	RMS	Max Torque
	2	q c2 = -178.4 o C 2 = 4.63 Nm 57.6% 34.7%
	3	q c3 = -221.4 o C 3 = 11.63 Nm 91.5% 46.4%
	4	q c4 = -176.9 o C 4 = 3.62 Nm 50.1% 33.1%
	5	q c5 = -224.2

o C 5 = 8.36 Nm 90.2% 44.1%

TABLE V .

 V Counterweight optimum configurations

Appendix 1