Dr Marc Dahan

Dr Laurentiu Racila

Experimental analysis of a translator based on a Wohlhart mechanism

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

INTRODUCTION

Many years ago, overconstrained mechanisms were analyzed only from theoretical point of view. Beside that, what are really missing are the industrial applications for these mechanisms. In 1857, Sarrus presented his six bars overconstrained mechanisms, capable of transforming circular motion in a linear one [START_REF] Sarrus | Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires ; et réciproquement[END_REF]. Since many authors have presented new and interesting overconstrained mechanisms. Excepting the Turbula mechanism developed by Paul Schatz [START_REF] Schatz | Rhytmusforschung und Technik[END_REF], overconstrained mechanisms has found very hardly place in the industrial field.

In 1987, Karl Wohlhart presents a new overconstrained mechanism, considered today as a generalisation of well known Bricard rectangular mechanism [START_REF] Wohlhart | A New 6R Space Mechanism[END_REF]. We call it the Wohlhart mechanism. When all links length is equal the mechanism became symmetric.

The translator presented in this study is based on this structure when the 6 links are the same length, but not in its "classical" position, when one element is fixed in space, but in a new spatial disposition when three non-successive joints are constrained to remain in a reference plane. Indeed, we have proved an important property of this mechanism: while the movements of the 3 joints in the fixed reference plane, the other three joints define a new plane, always parallel to the reference plane [START_REF] Racila | Bricard Mechanism Used as Translator[END_REF]. That's why we obtain a translator.

A prototype of the translator has been realized with the help of DG-Création Ltd, and a comparative study have been realized, putting in evidence the distance between the fixed reference plane and the mobile platform. The experimental analysis proves the rightness of the analytical calculus and the accuracy of the prototype

THEORETICAL APROACH

We consider Wohlhart mechanism with all links equals and noted with a and a twist angle α=π/2, but in a new spatial disposition. In this new spatial arrangement of the mechanism, we constrain three revolute joints O 1 , O 3 and O 5 to remain in the horizontal plane (fig. 1) and impose them to make a displacement on three line segments, to the centre of their circumscribed circle. The others three joints O 2 , O 4 and O 6 will remain always in a parallel plane with the first one [START_REF] Racila | Bricard Mechanism Used as Translator[END_REF]. The parallelism between the fixed plane and the platform can be proved using an analytical method or a geometrical one.

The geometrical approach gives this distance according to the input angle θ and output angle φ of Wohlhart mechanism [START_REF] Racila | Existance and kinematics of six bars spatial mechanisms[END_REF]:

2 sin 2 cos 3 4 1 2 sin 2 2 j q j ⋅ ⋅ - ⋅ ⋅ = a H (1)
We can write the expression of O 1 M 1 in the horizontal plane and O 1 O 6 O 5 triangle the next relation:

2 cos 2 3 1 1 q ⋅ = - = a b a M O ( 2
)
The relation [START_REF] Bricard | Mémoire sur la théorie de l'octaèdre articulé[END_REF] gives the dependence between the distance b and the angle θ.

The general input-output equation of Wohlhart symmetric mechanism is given by:

( ) ( ) 0 cos cos sin sin 2 sin cos cos cos 1 cos cos 2 2 2 = + ⋅ ⋅ ⋅ - - ⋅ + + + ⋅ ⋅ a a j q a j q a j q (3)
This relation can be particularised for a twist angle 2 p a = .

In this particular case we obtain the next input-output relation:

0 cos cos cos cos = + + ⋅ j q j q (4)
The analytical method gives the next relation for this distance between the two planes [START_REF] Racila | Existance and kinematics of six bars spatial mechanisms[END_REF]:

( ) ( ) q q j q q j 2 3 2 sin 1 cos sin 2 1 cos sin sin + + ⋅ ⋅ + ⋅ ⋅ ⋅ = a H ( 5
)
The relation ( 1) and ( 5) obtained using two different methods are equivalent.

We can write the relations (5) as:

q q q j q q j ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ = a H = (6) 2 sin 2 cos sin 4 2 cos sin sin 2 4 2 2 2 2 q q j q q j + ⋅ ⋅ ⋅ ⋅ ⋅ = a
We pass to the half angle and we add and subtract the term

q q j q q q q j + - + = a H = (7) = ˜˜˜¯Á Á Á Á Ë Ê + - 2 sin 2 cos 2 cos 2 sin 16 2 sin 1 2 sin 2 4 2 2 2 2 q q j j q q a
The input-output equation ( 4) can be written using the halfangles:

2 cos 4 1 2 cos 2 2 q j ⋅ = ( 8
)
Using eq. ( 8) the eq. ( 7) became:

˜˜˜¯Á Á Á Á Ë Ê + - = 2 sin 2 cos 2 sin 4 2 sin 1 2 sin 2 2 2 2 2 q q j q q a H = (9) ˙˙˙˚Í Í Í Í Î È + - - = 2 sin 1 2 cos 4 2 sin 1 2 sin 2 2 2 2 q q q q a
After some simplifications we obtain:

= ˙˙˙˚Í Í Í Í Î È - - = 2 cos 2 cos 1 3 1 1 2 sin 2 2 2 q q q a H (10) ˙˙˙˚Í Í Í Í Î È ˜˜˜¯Á Á Á Á Ë Ê - - = 1 2 cos 1 3 1 1 2 sin 2 2 q q a
Finally we obtain the same relation as the geometrical one:

= ˜˜˜¯Á Á Á Á Ë Ê - = 2 cos 4 1 1 2 sin 3 4 2 2 q q a H (11) 2 sin 2 cos 3 4 1 2 sin 2 2 j q j ⋅ ⋅ - ⋅ ⋅ = a
The results are identically for the two methods, proving the correctness of these two approaches.

Knowing the variation of the b distance (rel. 2) of actuated joints according to the input angle θ, we can plot the variation of the distance H between the horizontal plane and the mobile plane according to the distance b, for three values of the links length a (a = 40 mm, a = 80 mm, a = 120 mm), for a twist angle of α = π/2 (fig. 2).

Fig. 2 The variation ) (b f H = for a twist angle α = π/2 and for three links length, a=120, a=80 and a=40

EXPERIMENTAL RESULTS

A first model (fig. 3) was realised for a twist angle α = π/2 at the Applied Mechanical Laboratory in Besançon (France) with the help of DG-Création Ltd.

Fig. 3 The α = π/2 prototype To increase the precision, a second prototype was designed using a CAD software (fig. 4). ), because the different parts of the manipulator can be in collision. In fact, the manipulator gives the better precision in the second part of the variation curve, when the slope is the smallest.

The experimental tests realised with the first and the second prototype (fig. 5) allows as to tell that the experimental methods and the theoretical ones gives the same results, confirming the rightness of the calculus. Figure 6 show a detail of the actuated joint. This new structure of translator can be used in many industrial areas, as focusing plane in optical devices, as plate in rectifying machines or as an elevator, giving new perspectives for industrial applications based on overconstrained mechanisms.

Fig. 1

 1 Fig. 1 The translator in an intermediary positionWe consider an intermediary spatial position for the For need of clarity, we have not represented the O 3 joint (fig.1).The input parameter for the translator is considered the AO 1 distance, noted here with the b letter. The output parameter is the distance between the reference fixed plane O 1 O 3 O 5 and the mobile plane (the platform) O 2 O 4 O 6 .

Fig. 4

 4 Fig. 4 The CAD model of the second prototype We must note that the manipulator is not used in the two limit positions (θ=0° & b=0 and θ=120° & 3 / a b =), because the different parts of the manipulator can be in collision. In fact, the manipulator gives the better precision in the second part of the variation curve, when the slope is the smallest.The experimental tests realised with the first and the second prototype (fig.5) allows as to tell that the experimental methods and the theoretical ones gives the same results, confirming the rightness of the calculus. Figure6show a detail of the actuated joint.

Fig. 5 Fig. 6

 56 Fig. 5 The second prototype of 6R-translator