Keywords

1 Introduction

The industrial gripping devices usually use a reduced number of jaws 2 or 3 with equivalent dexterity of 40–90 % in comparison with the human hand having 5 fingers [1]. In [2], the gripping devices are classified in regard with the motion type of the jaws in translating, rotating (oscillating), planar motion and combinations of them. Konstantinov in [3] systematized the mechanisms used in gripping devices, which contains structural linkages, gears and screw-nut mechanisms. In [4] are shown some examples, which use in addition cam mechanisms in the structure of the gripping devices.

For all of these gripping devices an important industrial characteristic is the capability of centering precision. The centering precision is implicitly fulfilled in the case of the translational jaws and can be minimized for other motion types of the jaws. This aim is pursued by Qingsen in [5], and Brukher in [6], through proper designing of the gripping device mechanism provided by Pankin in [7] or through shaping the jaws. Recent researches develop special gripping devices used in soft robotics for performing gentle tasks [8] or for handling highly elastic objects [9]. For simple or complex anthropomorphic structures, underactuated grasping devices are used [10, 11], etc.

In [5], Qingsen presents a study of designing a grooved cam profile for increasing the centering precision. Simionescu in [12] shows a study of the centering precision of a conveniently chosen four-bar linkage used for acting a gripping device.

2 General Algorithm for Computing the Axial Centering Deviation

The centering is defined as the property of the gripping device to superpose the characteristic axis of the grasped object with his characteristic axis. The centering precision implies the computation of the axial centering deviation between the characteristic axis of the cylindrical grasped object and the characteristic axis of the grasping device. For this purpose it is considered the bisecting line of the prism in the top point M of the prism-jaw. The absolute axial centering deviation is the distance from the intersection of the bisecting line with the Ox-axis to the origin of the reference system axes xOy, identical with the characteristic axis of the grasping device (see Fig. 1).

Fig. 1
figure 1

Axial centering deviation

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The absolute axial centering deviation Δx computation, based on the notations in Fig. 1, follows by considering the vector equation:

$$ {\text{M}} =\Delta x + \frac{r}{\sin (\alpha /2)} \cdot e^{i(\vartheta - \beta - \alpha /2)} . $$
(1)

The coordinates x M and y M of the top point of the prism-jaw M and the pose angle ϑ of the coupler connecting element depend on the structure of the gripping mechanism. The prism angle of the jaw α (usually α = 120°) and the connection angle β of the coupler with the jaw are imposed by design conditions.

From the Eq. (1) can be computed the radius of the grasped object and the corresponding absolute axial centering deviation:

$$ \begin{array}{*{20}l} {r = y_{M} \cdot \frac{\sin (\alpha /2)}{\sin (\vartheta - \beta - \alpha /2)},} \hfill \\ {\Delta x = x_{M} + \frac{r}{\sin (\alpha /2)} \cdot \cos (\vartheta - \beta - \alpha /2).} \hfill \\ \end{array} $$
(2)

2.1 Gripping Device Using Parallelogram Linkages

The grapping device contains 2 parallelogram linkages symmetrically placed in respect with the axial direction of the structure and uses a symmetrical acting chain [13], as shown in Fig. 2. The motion of the coupler keeps parallel with the axial direction of the gripping device. The jaws prisms are fixed with the couplers.

Fig. 2
figure 2

Kinematic schema of the gripping device using parallel linkages

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The particularity of this parallelogram linkage consists in the constant value of the coupler angle:

$$ \vartheta (\varphi ) = \pi $$
(3)

and the same positional angle of the both cranks.

The coupler point M coordinates of the parallelogram linkage in the xOy coordinate system of the gripping device results from the vector equation:

$$ M = l_{01} + l_{1} + l_{4} \cdot e^{i \cdot \varphi } + (l_{3} + l_{31} ) \cdot e^{i \cdot \vartheta (\varphi )} , $$
(4)

and by considering the relationship (3) as:

$$ x_{M} = (l_{01} + l_{1} ) + l_{4} \cdot \cos \varphi - (l_{3} + l_{31} ),\quad y_{M} = l_{4} \cdot \sin \varphi . $$
(5)

2.2 Gripping Device Using Four-Bar Linkages

This type of gripping device (see Fig. 3) contains also 2 symmetrical four-bar linkages with coupler driven motion of the jaws and using symmetrical actuating chains [1, 14].

Fig. 3
figure 3

Kinematic schema of the grasping device using four-bar linkages with coupler jaws

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The positional analysis of the four-bar linkage, used for acting the coupler jaws, allows the computing of the coordinates of the top point M according to the kinematic parameters of the linkage. The vector equation of the four-bar linkage A0ABB0 in its own coordinate system, shown in Fig. 3, is:

$$ l_{1} + l_{4} \cdot e^{i \cdot \phi } + l_{5} \cdot e^{i \cdot (\vartheta + \delta )} = l_{6} \cdot e^{i \cdot \psi } . $$
(6)

By considering the complex conjugate equation of the vector Eq. (3) follows the coupler (5) positional angle ϑ depending on the rocker (4) angle φ:

$$ \vartheta (\phi ) = - \delta + \frac{{B_{1} (\phi ) + \sqrt {A_{1} (\phi ) + B_{1} (\phi ) - C_{1} (\phi )} }}{{A_{1} (\phi ) - C_{1} (\phi )}}. $$
(7)

with:

$$ A_{1} (\phi ) = 2l_{5} (l_{1} {\kern 1pt} + {\kern 1pt} l_{4} \cos \phi ),\quad B_{1} (\phi ) = 2l_{4} l_{5} \sin \phi ,\quad C_{1} (\phi ) = l_{1}^{2} {\kern 1pt} + {\kern 1pt} l_{4}^{2} {\kern 1pt} + {\kern 1pt} l_{5}^{2} {\kern 1pt} - {\kern 1pt} l_{6}^{2} {\kern 1pt} + 2l_{1} l_{4} \cos \phi . $$
(8)

The coordinates of the coupler point M of the four-bar linkage in the gripping device coordinate system xOy follows from the vector equation:

$$ M = l_{01} + l_{1} + l_{4} \cdot e^{i \cdot \phi } + l_{51} \cdot e^{i \cdot \vartheta (\phi )} , $$
(9)

in the form:

$$ \begin{aligned} x_{M} & = l_{01} + l_{1} + l_{4} \cdot \cos \phi + l_{51} \cdot \cos \vartheta (\phi ), \\ y_{M} & = l_{4} \cdot \sin \phi + l_{51} \cdot \sin \vartheta (\phi ). \\ \end{aligned} $$
(10)

3 Numerical Examples

The numerical examples show the variation of the absolute axial centering deviation in respect with the grasped object radius Δx(r) for the both grasping mechanism devices, presented in Sects. 2.1 and 2.2.

The considered radii of the grasped cylindrical object are in the range of 2.5–35 mm and the angle of the jaws prisms is chosen α = 120° for both examples.

3.1 Gripping Device Using Parallelogram Linkages

The geometrical parameters of the gripping device using parallelogram linkages are shown in Table 1.

Table 1 Geometrical parameters of the parallelogram linkage
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In Fig. 4a is represented the variation of the absolute axial centering deviation symmetrically distributed for the considered mechanism. The centering condition is fulfilled only for one radius of the grasped object (r = 26.66 mm), as expected.

Fig. 4
figure 4

Absolute axial centering deviation for parallelogram (a) and four-bar (b) linkage

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3.2 Gripping Device Using Four-Bar Linkage

The geometrical parameters of the gripping device using four-bar linkages are shown in Table 2 and the design assembly was patented in [15].

Table 2 Geometrical parameters of the parallelogram linkage
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Figure 4b shows the variation of the absolute axial centering deviation for the four-bar linkages. The centering condition is fulfilled for four radii of the grasped cylindrical object. For these radii the four-bar linkage was computed by using four pose synthesis in [1, 14].

4 Conclusions

The paper presents a general algorithm for computing the absolute axial centering deviation, as theoretical precision parameter. Some known gripping devices were considered and numerically analyzed to validate the proposed algorithm.