Six-Link Articulated Mechanisms
Abstract Generated abstract
This paper examines exceptional six-link spatial closed chains with revolute joints, which are generally rigid by the standard mobility count but can become mechanisms under special geometric conditions. It classifies known examples into a first group with two pencils of adjacent hinge axes, then presents new model-based variants with symmetric opposite links and with a plane of symmetry, specifying parameter relations among link lengths, hinge-axis angles, and axial offsets. For the symmetric-plane group, the paper gives a geometric justification by reducing motion to the continuous intersection of two hinge axes whose swept surfaces are hyperboloids or cones. It concludes that, except in limiting tangency cases, a symmetric six-link configuration whose symmetry plane passes through the relevant axes can possess mobility.
Full Text
SIX-LINK SPATIAL HINGED MECHANISMS
A. V. VERKHOVSKII
Professor, Doctor of Technical Sciences
Introduction
It is known that the general form of a spatial hinged mechanism composed as a simple closed kinematic chain is a seven-link mechanism.
A six-link hinged closed chain composed solely of revolute pairs is, in the general case, a rigid system. This can be seen from the structural formula for a mechanism
\[ 6(n-1)-5p=W, \]
where \(n\) is the number of links, including the frame,
\(p\) is the number of revolute kinematic pairs (hinges),
\(W\) is the number of degrees of mobility of the mechanism.
Taking \(n=6\) and \(p=6\), we obtain
\[ 6(6-1)-5.6=0, \]
i.e., the number of degrees of mobility of the system is found to be zero.
This means that, in the general case, a six-link hinged system (Fig. 1) cannot be a mechanism.
At the same time it is known that, when certain special conditions are observed, a six-link hinged system composed as a simple closed kinematic chain acquires one degree of mobility, i.e., becomes a mechanism.
In this work several new six-link hinged mechanisms are considered.
Fig. 1
Mechanisms of Group I
At present, several examples of six-link hinged mechanisms are known in the literature, namely:
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A six-link mechanism in which the axes of adjacent hinges form two bundles of intersecting axes, three axes in each. Described in the article by Rittershaus [1].
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A six-link mechanism in which the axes of adjacent hinges form two bundles of parallel axes, three axes in each. Described there as well.
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A six-link mechanism in which the axes of three adjacent hinges form one bundle of parallel axes, and the remaining three axes form a bundle of intersecting axes. Described by Prof. N. G. Bruyevich [2].
Let us note that all three mechanisms can be defined as mechanisms with two bundles of adjacent axes, three axes in each, regardless of whether the axes in a bundle intersect or are parallel. A bundle with parallel axes may be regarded as the limiting case of a bundle with intersecting axes, in which the point of intersection of the axes is removed to infinity.
Thus, all three mechanisms can be combined into one group, which we shall call group I.
Let us also note that the mechanisms described may be regarded as doubled four-link planar and spherical mechanisms. To obtain a six-link mechanism, the doubling process must be carried out as follows. The initial four-link mechanisms should be opened at some hinges, and the freed links should be rigidly connected pairwise with one another.
In doing this it is necessary to ensure that, when two planar mechanisms are doubled, all hinge axes do not become parallel, and that, when two spherical mechanisms are doubled, they do not intersect at one point.
The purpose of the present work is to give several new examples of six-link spatial hinged mechanisms and to provide a geometric justification for some of them.
For convenience of description, let us introduce the following notation for the parameters of the mechanism:
by the letter \(r\) we shall denote the shortest distance between the hinge axes of a link (the length of the link);
by the letter \(\alpha\)—the angle between the hinge axes of a link;
by the letter \(\lambda\)—the segment on the hinge axis enclosed between the ends of the links adjoining the hinge.
To the letters \(r\) and \(\alpha\) we shall assign subscripts corresponding to the number of the link. To the letter \(\lambda\) we shall assign a double subscript corresponding to the numbers of the links adjoining the given hinge.
Mechanisms of group II
As indicated above, to the first group we assigned mechanisms having two bundles of axes of adjacently arranged hinges, three axes in each.
To the second group we shall assign mechanisms having opposite links symmetric.
On the basis of constructing models, it was possible to discover several possible mechanisms of this group, which are brought together in Table 1.
The table indicates the presence in the mechanisms of bundles of intersecting axes of adjacent hinges and the number of axes in them. This is important when transforming a mechanism by the method of Prof. Artobolevskii. Applying this method, one may replace hinges with intersecting axes by multi-axis kinematic pairs, while discarding the intermediate links.
Bundles that are formed by the axes of hinges not arranged adjacent to one another are not indicated in the table.
In Table 1, for the third and fourth variants, 4 and 6 bundles of axes of two each are indicated, whereas the mechanism has only 6 axes in all. This is explained by the fact that some axes participate simultaneously in two bundles.
Let us note that the fifth and sixth variants of the group under consideration are at the same time special cases of a mechanism of group I, since they have two bundles of axes of three each.
Let us also draw attention to one mechanism, which is placed in Table 1 as the second special case of the first variant. This mechanism is characterized by equality of the lengths of all links (\(r\)) and of the angles between the axes.
hinges \((\alpha)\). In this mechanism the axes of the hinges form two bundles, three in each, intersecting successively through one. At the same time
Table 1. Table of six-link mechanisms of group II
Conditional diagram of the mechanism
| Variant | Conditions for existence of the mechanism of the given variant | Number of model constructed | Number of degrees of freedom | Additional conditions of particular cases: first particular case | Number of model constructed | Additional conditions of particular cases: second particular case | Additional conditions of particular cases: third particular case |
|---|---|---|---|---|---|---|---|
| I | \(r_1=r_4,\ r_2=r_5,\ r_3=r_6\) \(\alpha_1=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_3=\alpha_6\) \(\lambda_{12}=\lambda_{45};\ \lambda_{34}=\lambda_{56};\ \lambda_{35}=\lambda_{46}=0\) * |
— | — | \(r_1=r_2=r_4=r_5\) \(\alpha_1,\alpha_2=\alpha_4=\alpha_5\) |
— | \(r_1=r_2=r_4=r_5=r_6\) \(\alpha_1,\alpha_2=\alpha_3;\ \alpha_4=\alpha_5=\alpha_6\) * |
\(r_2=\infty,\ r_5=\infty\) \(r_5=\infty;\ r_6=\infty\) |
| II | \(r_1=r_4=0;\ r_2=r_5;\ r_3=r_6\) \(\alpha_1=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_3=\alpha_6\) \(\lambda_{12}=\lambda_{45};\ \lambda_{34}=\lambda_{56};\ \lambda_{35}=\lambda_{46}=\lambda_{61}=0\) |
2 2 |
\(\lambda_{12}=\lambda_{34}=\lambda_{45}=\lambda_{61}=0\) \(\lambda_{23}=\lambda_{56}\ne0\) |
2 2 |
|||
| III | \(r_1=r_4=0;\ r_2=r_5;\ r_3=r_6\) \(\alpha_1=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_3=\alpha_6\) \(\lambda_{12}=\lambda_{43};\ \lambda_{25}=\lambda_{36};\ \lambda_{34}=\lambda_{51}\) |
2 2 2 2 |
— | ||||
| IV | \(r_1=r_2=r_4=r_5=0;\ r_3=r_6\) \(\alpha_3=\alpha_4=0;\ \alpha_1=\alpha_4;\ \alpha_2=\alpha_5\) \(\lambda_{12}=\lambda_{45};\ \lambda_{23}=\lambda_{56};\ \lambda_{34}=\lambda_{61}\) |
2 2 2 2 2 |
2 2 |
||||
| V | \(r_1=r_4=0;\ r_2=r_5=0;\ r_3=r_6\) \(\alpha_1=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_3=\alpha_6\) \(\lambda_{12}=\lambda_{45}=0;\ \lambda_{23}=\lambda_{56};\ \lambda_{34}=\lambda_{61}\) |
3 3 |
— | ||||
| VI | \(r_1=r_4;\ r_2=r_5;\ r_3=r_6\) \(\alpha_1=\alpha_4=0;\ \alpha_2=\alpha_5;\ \alpha_3=\alpha_6\) \(\lambda_{12}=\lambda_{45};\ \lambda_{23}=\lambda_{60};\ \lambda_{34}=\lambda_{61}\) |
— | 3 3 |
Note to Table 1. Models of the mechanisms marked with an asterisk were constructed by the author in 1923 and added to the collection of mechanisms of the Museum of Applied Mechanics of the Tomsk Technological Institute. In 1924, a report on these mechanisms was delivered by the author at a meeting of the Council of the Tomsk Institute of Applied [[unclear: continuation cut off]].
the mechanism has three planes of symmetry, and for this reason it will also be included in group III, to the consideration of which we now proceed.
Mechanisms of group III
Group III includes mechanisms having a plane of symmetry. The most characteristic mechanism of this group is shown schematically in Fig. 2.
The mechanisms of the group under consideration, whose existence was established by constructing models, are summarized in Table 2.
In Table 2 we note the particular case of the second variant. In this variant of the mechanism the lengths of all links \((r)\) and the angles between the axes of the hinges \((\alpha)\) are equal. The mechanism has three planes of symmetry and two bun-
Table of six-link mechanisms of group III
Table 2
Schematic diagram of the mechanism
| Variant | Conditions for existence of the mechanism | Number of intersections | Number of positions | Additional conditions of special cases: special case | Number of intersections | Number of positions |
|---|---|---|---|---|---|---|
| I | $r_3=r_4;\ r_2=r_5;\ r_1=r_6$ $\alpha_3=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_1=\alpha_6$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
— | — | |||
| II | $r_3=r_4;\ r_2=r_5;\ r_1=r_6$ $\alpha_3=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_1=\alpha_6$ $\lambda_{12}=\lambda_{23}=\lambda_{34}=\lambda_{45}=\lambda_{56}=\lambda_{61}=0$ |
— | — | $r_1=r_2=r_3=r_4=r_5=r_6$ $\alpha_1=\alpha_2=\alpha_3=\alpha_4=\alpha_5=\alpha_6$ |
||
| III | $r_3=r_4=0;\ r_2=r_5;\ r_1=r_6$ $\alpha_3=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_1=\alpha_6$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
3 | — | |||
| IV | $r_3=r_4;\ r_2=r_5;\ r_1=r_6$ $\alpha_3=\alpha_4=0;\ \alpha_2=\alpha_5;\ \alpha_1=\alpha_6$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
— | 3 | |||
| V | $r_3=r_4;\ r_2=r_5=0;\ r_1=r_6$ $\alpha_3=\alpha_4;\ \alpha_2=\alpha_5;\ \alpha_1=\alpha_6$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
2 2 |
$r_2=r_5=0,\ r_1=r_3=r_4=r_6$ $\alpha_2=\alpha_5,\ \alpha_1=\alpha_3=\alpha_4=\alpha_6$ |
2 2 |
||
| VI | $r_3=r_4;\ r_2=r_5;\ r_1=r_6$ $\alpha_3=\alpha_4;\ \alpha_2=\alpha_5=0;\ \alpha_1=\alpha_6$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
2 2 |
$r_2=r_5;\ r_3=r_4=r_1=r_6$ $\alpha_2=\alpha_5=0;\ \alpha_1=\alpha_3=\alpha_4=\alpha_6$ |
2 2 |
||
| VII | $r_1=r_6=r_3=r_4=0;\ r_2=r_5$ $\alpha_1=\alpha_6;\ \alpha_3=\alpha_4;\ \alpha_2=\alpha_5$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
3 3 |
$\alpha_2=\alpha_5=0$ | 3 3 |
2 2 |
|
| VIII | $r_1=r_6;\ r_3=r_4;\ r_2=r_5$ $\alpha_1=\alpha_6=\alpha_3=\alpha_4=0;\ \alpha_2=\alpha_5$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
— | 3 3 |
$r_2=r_5=0$ | 2 2 |
3 3 |
| IX | $r_1=r_6=0;\ r_3=r_4;\ r_2=r_5$ $\alpha_3=\alpha_4=0;\ \alpha_1=\alpha_6;\ \alpha_2=\alpha_5$ $\lambda_{23}=\lambda_{45};\ \lambda_{12}=\lambda_{56};\ \lambda_{34}=\lambda_{16}=0$ |
3 | 3 |
... axes, three in each. In each pencil the axes of the hinges alternate successively one after another. This mechanism, as indicated above, was also included in Table 1, for the reason that its opposite links are symmetric.
Let us note that the seventh, eighth, and ninth variants of the group under consideration are at the same time special cases of a mechanism of group I, since they have two pencils of hinge axes, with three axes in each pencil.
Geometric substantiation of the mechanisms of the third group
As already indicated, the mechanisms of groups II and III, placed in Tables 1 and 2, were obtained by constructing models.
In the present paper we confine ourselves to the geometric investigation only of the mechanisms of group III.
The mechanisms we have assigned to group III are those having a plane of symmetry. The most characteristic mechanism of this group is shown schematically in Fig. 2.
Fig. 2
To carry out the investigation, let us divide the mechanism by its plane of symmetry, i.e., along the axes \(A\) and \(D\), into two parts. Let us place one of these parts in a system of rectangular axes as shown in Fig. 3, i.e., let the link \(r_2\) coincide with the axis \(X\), and the axis \(B\) of the mechanism with the axis \(Y\).
The group of mechanisms under consideration has a plane of symmetry passing through the axes \(A\) and \(D\).
It follows from this that, for the geometric substantiation of the mechanisms of this group, it is sufficient to find such conditions under which the axes \(A\) and \(D\) would continuously intersect each other during the motion of the mechanism.
Fig. 3
From Fig. 3 it is not difficult to see that the axes \(A\) and \(D\), upon rotation of the links \(r_1\) and \(r_3\), will describe surfaces of hyperboloids, and in the special case, when \(r_1 = 0\) or \(r_3 = 0\), surfaces of cones.
By choosing the length of the link \(r_2\) and the angle \(\alpha_2\), as well as \(\lambda_{12}\) and \(\lambda_{23}\), one can always achieve that the surfaces of the hyperboloids or cones under consideration will intersect one another.
The line of their intersection will represent the locus of points of successive and continuous intersection of the axes $A$ and $D$ in the course of their motion.
Thus, the condition for the existence of mechanisms of the third group is the presence of an intersection of the indicated hyperbolic or conical surfaces.
Practically any mechanism of the group under consideration can easily be obtained in the following way.
One should construct the configuration of the desired (symmetric) mechanism in one arbitrary position. By virtue of the symmetry of the constructed configuration, the axes $A$ and $D$ will intersect each other at some point in space.
If, by chance, the point of intersection of the axes $A$ and $D$ is not a point of tangency of the hyperbolic or conical surfaces described by these axes (which is unlikely), then it will be one of the points of the line of intersection of the indicated surfaces. And this means that the constructed configuration will be movable, i.e., will be a mechanism.
Thus, if we do not have the limiting possible case of tangency of the surfaces, then it may be said that any constructed six-link symmetric configuration in which the plane of symmetry passes through the axes will be a mechanism, with the exception of the indicated case of tangency.
LITERATURE
- Bittreshans.—Zur heutigen Schule der Kinematik. Der Civilingenieur, 1875.
- Bruyevich N. G.—Kinematics of spatial mechanisms. Proceedings of the Air Force Academy, collection No. 22, 1937.