# Experimental analysis of a translator based on a Wohlhart mechanism

Marc Dahan, Laurentiu Racila

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► To cite this version:

Marc Dahan, Laurentiu Racila. Experimental analysis of a translator based on a Wohlhart mechanism. 9th ASME Engineering Systems Conference, Jul 2008, Haifa, Israel. pp.527-530, 10.1115/ESDA2008-59157. hal-00459049

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# EXPERIMENTAL ANALYSIS OF A TRANSLATOR BASED ON A WOHLHART MECHANISM

Dr. Marc DAHAN

CNRS / Franche-Comté University, France

Dr. Laurentiu RACILA

ESIREM / Bourgogne University, France

## ABSTRACT

The aim of this paper is to present a theoretical and experimental analysis of 6R-translator based on Wohlhart 6R overconstrained mechanism. The variable distance between the fixed plane and the platform of the 6R-translator is obtained using a theoretical approach and then validated experimentally.

## 1. 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 [7]. Since many authors have presented new and interesting overconstrained mechanisms. Excepting the Turbula mechanism developed by Paul Schatz [8], 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 [10]. 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-succcessive 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 [6]. 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.

## 2. 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 [6].

<figure><img src="image_3.png" /><figcaption>Fig. 1 The translator in an intermediary position</figcaption></figure>

We consider an intermediary spatial position for the translator. 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 $O_1O_3O_5$ and the mobile plane (the platform) $O_2O_4O_6$.
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 $\theta$ and output angle $\varphi$ of Wohlhart mechanism
[5]:

$$
H = a \cdot \sin \frac{\varphi}{2} \cdot \sqrt{1 - \frac{4}{3} \cdot \cos^2 \frac{\theta}{2} \cdot \sin^2 \frac{\varphi}{2}} \quad (1)
$$

We can write the expression of $O_1M_1$ in the horizontal
plane and $O_1O_6O_5$ triangle the next relation:

$$
O_1 M_1 = a - \frac{b\sqrt{3}}{2} = a \cdot \cos \frac{\theta}{2} \quad (2)
$$

The relation (2) gives the dependence between the distance
$b$ and the angle $\theta$.

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

$$
\cos \theta \cdot \cos \varphi \cdot (1 + \cos^2 \alpha) + (\cos \theta + \cos \varphi) \cdot \sin^2 \alpha - \\ - 2 \cdot \sin \theta \cdot \sin \varphi \cdot \cos \alpha + \cos^2 \alpha = 0 \quad (3)
$$

This relation can be particularised for a twist angle $\alpha = \frac{\pi}{2}$.
In this particular case we obtain the next input-output relation:

$$
\cos \theta \cdot \cos \varphi + \cos \theta + \cos \varphi = 0 \quad (4)
$$

The analytical method gives the next relation for this
distance between the two planes [5]:

$$
H = \frac{a \cdot \sin \varphi \cdot \sin \theta \cdot (\cos \theta + 1)}{\sqrt{2 \cdot \sin^2 \varphi \cdot (\cos \theta + 1)^3 + \sin^2 \theta}} \quad (5)
$$

The relation (1) and (5) obtained using two different
methods are equivalent.

We can write the relations (5) as:

$$
\begin{align}
H &= \frac{2a \cdot \sin \varphi \cdot \sin \theta \cdot \cos^2 \frac{\theta}{2}}{\sqrt{16 \cdot \sin^2 \varphi \cdot \cos^6 \frac{\theta}{2} + 4 \cdot \sin^2 \frac{\theta}{2} \cdot \cos^2 \frac{\theta}{2}}} = \tag{6} \\
  &= a \cdot \sqrt{\frac{\sin^2 \varphi \cdot \sin^2 \theta \cdot \cos^2 \frac{\theta}{2}}{4 \cdot \sin^2 \varphi \cdot \cos^4 \frac{\theta}{2} + \sin^2 \frac{\theta}{2}}} \nonumber
\end{align}
$$

We pass to the half angle and we add and subtract the term
$\sin^4 \frac{\theta}{2}$:

$$
\begin{align}
H &= a \sqrt{\frac{4 \sin^2 \varphi \cos^4 \frac{\theta}{2} \sin^2 \frac{\theta}{2} + \sin^4 \frac{\theta}{2} - \sin^4 \frac{\theta}{2}}{4 \sin^2 \varphi \cos^4 \frac{\theta}{2} + \sin^2 \frac{\theta}{2}}} = \tag{7} \\
  &= a \sqrt{\sin^2 \frac{\theta}{2} \left( 1 - \frac{\sin^2 \frac{\theta}{2}}{16 \sin^2 \frac{\varphi}{2} \cos^2 \frac{\varphi}{2} \cos^4 \frac{\theta}{2} + \sin^2 \frac{\theta}{2}} \right)} \nonumber
\end{align}
$$

The input-output equation (4) can be written using the half-
angles:

$$
\cos^2 \frac{\varphi}{2} = \frac{1}{4 \cdot \cos^2 \frac{\theta}{2}} \quad (8)
$$

Using eq. (8) the eq. (7) became:

$$
\begin{align}
H &= a \sqrt{\sin^2 \frac{\theta}{2} \left( 1 - \frac{\sin^2 \frac{\theta}{2}}{4 \sin^2 \frac{\varphi}{2} \cos^2 \frac{\varphi}{2} + \sin^2 \frac{\theta}{2}} \right)} = \tag{9} \\
  &= a \sqrt{\sin^2 \frac{\theta}{2} \left[ 1 - \frac{\sin^2 \frac{\theta}{2}}{4 \cos^2 \frac{\theta}{2} - 1 + \sin^2 \frac{\theta}{2}} \right]} \nonumber
\end{align}
$$

After some simplifications we obtain:
$$ 
\begin{aligned}
H &= a \sqrt{\sin^2 \frac{\theta}{2} \left[ 1 - \frac{1}{3} \frac{1 - \cos^2 \frac{\theta}{2}}{\cos^2 \frac{\theta}{2}} \right]} = \\
  &= a \sqrt{\sin^2 \frac{\theta}{2} \left[ 1 - \frac{1}{3} \left( \frac{1}{\cos^2 \frac{\theta}{2} - 1} \right) \right]}
\end{aligned} 
\quad (10) $$

Finally we obtain the same relation as the geometrical one:

$$ 
\begin{aligned}
H &= a \sqrt{\frac{4}{3} \sin^2 \frac{\theta}{2} \left( 1 - \frac{1}{4 \cos^2 \frac{\theta}{2}} \right)} = \\
  &= a \cdot \sin \frac{\varphi}{2} \sqrt{1 - \frac{4}{3} \cos^2 \frac{\theta}{2} \sin^2 \frac{\varphi}{2}}
\end{aligned} 
\quad (11) $$

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 $\theta$, 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 $\alpha = \pi/2$ (fig. 2).

<figure><img src="image_4.png" /><figcaption>Fig. 2 The variation $H = f(b)$ for a twist angle $\alpha = \pi/2$ and for three links length, a=120, a=80 and a=40</figcaption></figure>

### 3. EXPERIMENTAL RESULTS

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

<figure><img src="image_5.png" /><figcaption>Fig. 3 The $\alpha = \pi/2$ prototype</figcaption></figure>

To increase the precision, a second prototype was designed using a CAD software (fig. 4).

<figure><img src="image_6.png" /><figcaption>Fig. 4 The CAD model of the second prototype</figcaption></figure>

We must note that the manipulator is not used in the two limit positions ($\theta=0°$ & $b=0$ and $\theta=120°$ & $b = a/\sqrt{3}$), 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.
<figure><img src="image_7.png" /><figcaption>**Fig. 5** The second prototype of 6R-translator</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6** The actuated joint of the second prototype</figcaption></figure>

## 4. CONCLUSIONS

The paper shows the equivalence between the theoretical (geometrical and analytical) and the experimental methods for the 6R-translator based on Wohlhart symmetric mechanism.

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.

## 5. REFERENCES

[1] Baker J.E., 1980, "An Analysis of the Bricard Linkages", Mech. Mach. Theory, vol. 15, pages 267-286.

[2] Bricard R., 1897, "Mémoire sur la théorie de l'octaèdre articulé", Journal de Mathématiques pures et appliquées, Liouville 3, pages 113-148.

[3] Bricard R., 1927, "Leçons de Cinématique", Tome II - Cinématique Appliquée, Gauthier-Villars, Paris.

[4] Dietmaier P., 1995, "A new 6R space mechanism", Proceedings of the 9th IFTOMM World Congress, vol. 1, pages 52-56, Milano, Italy.

[5] Racila L., 2006, "Existence and kinematics of six bars spatial mechanisms", Ph.D, Franche-Comte University.

[6] Racila L., and Dahan M., 2007, "Bricard Mechanism Used as Translator", In Proceedings of the 12th IFTOMM World Congress, vol. 3, pages 337-341, Besançon, France

[7] Sarrus P-F., 1853, "Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires; et réciproquement", Comptes Rendus des Séances de l'Académie des Sciences, Vol. 36, p. 1036-1038.

[8] Schatz P., 1975, "Rhytmusforschung und Technik", Verlag freies Geistesleben, Stuttgart.

[9] Waldron K.J., 1979, "Overconstrained Linkages", Environment and Planning B, Vol. 6, pages 393-402.

[10] Wohlhart K., 1987, "A New 6R Space Mechanism", Proceedings of the 7th IFTOMM World Congress, vol. 1, pages 193-198, Sevilla.

[11] Wohlhart K., 1993, "The Two Types of the Orthogonal Bricard Linkage", In Mech. Mach. Theory, Vol. 28, pages 809-817.