A Six-Link Spatial Mechanism with an Axis of Symmetry

V. M. Avraamov

Introduction

Prof. A. V. Verkhovsky, in the article “Six-Link Spatial Hinged Mechanisms”¹, describes several new six-link spatial hinged mechanisms discovered by him on the basis of constructing models.

Prof. A. V. Verkhovsky divides the mechanisms he discovered into two groups. To one group he assigns mechanisms having planes of symmetry; to the other—mechanisms in which “opposite links are symmetrical.”

For mechanisms with a plane of symmetry, the article gives a geometric substantiation of the conditions for their existence.

For mechanisms of the second group, however, there is only an indication that in them “opposite links are symmetrical,” and a table is given of the conditions for the existence of six variants of mechanisms, compiled on the basis of constructing models. No theoretical substantiation of the possibility of the existence of mechanisms of this group is given in the work. At the same time, as it seems to us, the basic, general case of a mechanism in which all the parameters of the mechanism would be different and not equal to zero does not appear in the work at all.

The purpose of the present work is to prove the possibility of the existence of the general case of a mechanism of this group, and consequently also of its particular variants discovered by Prof. A. V. Verkhovsky on models.

A Six-Link Spatial Hinged Mechanism with an Axis of Symmetry (General Case)

Let us consider the question of the possibility of the existence of a spatial six-link hinged mechanism with an axis of symmetry, i.e., a mechanism whose links, during motion, retain a position pairwise symmetrical with respect to some axis.

A conventional diagram of such a mechanism is shown in Fig. 1, where \(r\) denotes the lengths of the links, i.e., the shortest distances between the axes of the hinges of a link; \(\alpha\) (not indicated in the figure) denotes the angles between the axes of the hinges of a link; and \(\lambda\) denotes the segments on the axes of the hinges measuring the shortest distances between the axes of two adjacent links.

Let us dwell on the characterization of the position of two bodies symmetrical with respect to an axis.

Two bodies have an axis of symmetry if

1) they are geometrically identical, and
2) their corresponding points are situated on a common perpendicular to the axis (of symmetry) and are equidistant from it.

From this definition of symmetry there follows, as a consequence, a number of geometric properties of the symmetrical arrangement of bodies (Fig. 2).

a) the position of the second body is obtained by rotating the first about the axis of symmetry through \(180^\circ\);

b) the projection of the bodies onto a plane parallel to the axis of symmetry represents a plane figure with an axis of symmetry, while their projection onto a plane perpendicular to the axis of symmetry is a figure with a center of symmetry;

c) through two pairs of symmetrical points (for example, \(A_1, A_2\) and \(B_1, B_2\)) one can always draw two pairs (and only two) of symmetrical lines \((A_1B_1, A_2B_2\) and \(A_1B_2, A_2B_1)\), which is evident from the drawing. Hence, in turn, it follows:

d) to a figure that includes the geometric elements of both symmetrical bodies there always corresponds a symmetrical figure (for example, to the figure \(A_1B_1A_2C_2A_1\) there corresponds the figure \(A_2B_2A_1C_1A_2\)) or

Fig. 1

Fig. 2

e) the relative position of two arbitrary (not symmetrical) geometric elements (for example, the lines \(A_1B_1\) and \(A_2C_2\)) and the relative position of the elements symmetrical to them (the lines \(A_2B_2\) and \(A_1C_1\)) are geometrically identical; when the position of the former changes, the latter change their position in an identical manner. This property is especially important for what follows.

From the very definition of the position of bodies symmetrical with respect to an axis it follows that the symmetrical relative motion of two bodies has at least two constraints—the impossibility of relative translational motion along the axis of symmetry and of relative rotation about it; consequently, the number of degrees of freedom of this motion will be no more than four.

We shall show that in the general case the relative motion of two bodies symmetrical with respect to an axis indeed possesses exactly four degrees of freedom.

As is known, the position of a body in space is determined by six independent generalized coordinates. For our case it is expedient to associate these coordinates with three points of the body and to express them in a cylindrical coordinate system, aligning in it the axis of the cylinder with the axis of symmetry of the bodies.

Then the position of the first body is determined (Fig. 2) by three coordinates of one of its points (for example, \(A_1\))—\(r_{A1}\), \(\varphi_{A1}\), and \(z_{A1}\), by two coordinates of another (\(B_1\))—\(r_{B1}\), \(z_{B1}\), and by one coordinate of a third point (\(C_1\))—\(r_{C1}\).
And, correspondingly, for the second body:

\[ r_{A2},\ \varphi_{A1},\ z_{A2},\ r_{B2},\ z_{B2},\ r_{C2} \]

The condition of a symmetric arrangement of the bodies imposes functional relations (constraint conditions) on some coordinates and turns them from independent into dependent ones. Thus, the condition that the symmetric points of the bodies lie on the common perpendicular to the axis of symmetry imposes the conditions:

\[ z_{A1}=z_{A2}, \qquad z_{B1}=z_{B2}, \qquad \varphi_{A1}=\varphi_{A2}, \]

and the conditions of their equal distance from the axis of symmetry are the conditions:

\[ r_{A1}=r_{A2}, \qquad r_{B1}=r_{B2}, \qquad r_{C1}=r_{C2}. \]

Thus, on the 12 coordinates of two bodies the condition of symmetry imposes 6 constraint conditions; consequently, only six independent coordinates remain.

Since we are interested in the relative position of the bodies, while the coordinates \(z_{A1}\) and \(\varphi_{A1}\) characterize only the position of the entire system, they, as not reflecting the relative position of the bodies, must be discarded, and the coordinate \(z_{B1}\) must be modified into \(z_{B1A1}\), i.e., it must be expressed through the relative distance of the points \(B_1\) and \(A_1\).

Thus, the independent coordinates determining the relative position of two symmetric bodies through the position of the axis of symmetry will be only four coordinates (parameters):

\[ r_A,\ r_B,\ r_C \ \text{and}\ z_{BA} \]

But since the number of degrees of freedom of a system is numerically equal to the number of independent coordinates determining its position, the number of degrees of freedom of two bodies performing symmetric relative motion with respect to an axis will be equal to four, as was required to prove.

The simplest independent motions compatible with the constraint conditions will obviously be two translational motions along two axes perpendicular to the axis of symmetry, and two rotational motions about them.

To realize this general case of symmetric relative motion with four degrees of freedom of two bodies, one may use, for example, a system of three plane symmetric six-link hinge mechanisms having a common axis of symmetry and capable of moving independently along and about it (Fig. 3).

Each of the mechanisms is attached to symmetric points of the bodies by means of ball-and-socket joints, thereby achieving a symmetric arrangement, relative to the axis, of three pairs of points of the bodies.

Since each of the mechanisms, moreover, can move independently along and about the axis of symmetry \(0—0\), all three mechanisms together impose on the motion of the bodies all four necessary restrictions corresponding to the conditions of symmetry and, in this way, ensure the required motion of the two bodies relative to the axis with four degrees of freedom.

The resulting system has 20 links and 43 axes and is a mechanism with four degrees of mobility.

From the general case of symmetric motion with four degrees of freedom, by imposing constraints on it, one can obtain symmetric motions with smaller numbers of degrees of freedom.

To obtain symmetric motion with one degree of freedom, it is necessary, in the general case of symmetric motion, to introduce three constraints, three constraint conditions. These constraints may be obtained, for example, by attaching, at arbitrary points of the symmetric bodies, a spatial hinged dyad (with arbitrary parameters), since it is known that the attachment of such a dyad introduces into the system 12 new independent coordinates and 15 constraint conditions, i.e., adds to the system three additional constraint conditions.

In Fig. 3 the cylindrical joints of this dyad are denoted by the letters \(A_1\), \(D_1\), and \(C_2\). As a result of this attachment, the former 20-link mechanism is transformed into a 22-link, 46-axis mechanism with 1 degree of freedom.

According to the property noted earlier (d) of the symmetric arrangement of bodies, it follows that if at the points \(A_2\) and \(C_1\), symmetric to the points \(A_1\) and \(C_2\), new joints are fixed, symmetric to the corresponding joints \(A_1\) and \(C_2\) of the dyad \(A_1 D_1 C_2\), then their relative motion will be identical to the motion of the joints \(A_1\) and \(C_2\). Consequently, the attachment to them of a new dyad \(A_2 D_2 C_1\), identical to the dyad \(A_1 D_1 C_2\), will not affect the motion of the bodies, since the constraint conditions introduced by this dyad are identical with those already present in the mechanism. In view of the fact that these constraints do not affect the motion of the mechanism, they may be characterized as passive.

Fig. 3

As a result of attaching both dyads, the mechanism is transformed into a 24-link, 49-axis mechanism, in which two main links (bodies) and two dyads form a closed contour. Detaching from this mechanism the system of planar mechanisms transfers the constraints of the dyad \(A_2 D_2 C_1\) from passive to active and transforms the remaining system into a closed spatial six-link hinged kinematic chain, the relative motions of whose links are pairwise symmetric with respect to a common axis, i.e., a spatial six-link hinged mechanism with an axis of symmetry is obtained, hereafter called a six-axis spatial mechanism with an axis of symmetry (since what is characteristic is not the number of links, but the number of axes in it).

A schematic diagram of such a mechanism was given in Fig. 1.

Since the points of attachment of the dyad and the directions of the axes of the joints, as well as the parameters of the dyad, were arbitrary, the total number of independent parameters determining the mechanism will be equal to nine: three parameters \(r\), \(\lambda\), and \(\alpha\) for each link of the symmetric part of the chain.

A brief résumé of the proof carried out for the existence of a six-axis mechanism with an axis of symmetry is as follows.

On the basis of the properties of motion symmetric with respect to an axis:

1) the motion, symmetric with respect to an axis, of two links is a motion with four degrees of freedom;

2) the relative motion of a pair of segments connected with symmetrically moving links and the relative motion of a second pair of segments, symmetric to the first, are geometrically identical; it follows that:

a) the attachment to the links performing motion symmetric with respect to the axis of a spatial hinged dyad reduces the number of degrees of freedom of the symmetric motion of the links to one;

b) the attachment to the links of a second, identical dyad, symmetric with respect to the axis, produces no additional constraint on the freedom of motion (on the basis of item 2), but only closes the kinematic chain and transforms the system into a mechanism.

Fig. 4

Fig. 4 shows a photograph of the most characteristic six-link spatial mechanism with an axis of symmetry; in it all like-named parameters of the symmetric links are different and are not equal to zero. For the given particular values of the parameters the mechanism proved to be a double-crank mechanism.

This mechanism was demonstrated in February 1950 at a scientific seminar of the Mechanical Faculty of the Tomsk Polytechnic Institute and was transferred to the museum of the Department of Applied Mechanics of TPI.

Conclusion

1. A six-link spatial mechanism with an axis of symmetry has been constructed and theoretically substantiated; it consists of two geometrically identical, in the general case three-link, hinged chains, the links of which, in pairs, perform relative motions symmetric with respect to the common axis.

The mechanism is characterized by nine independent parameters, three parameters \(r\), \(\lambda\), and \(\alpha\) for each link of the identical three-link chain.

1. It has been shown that the kinematic chains of group II, discovered by Prof. A. V. Verkhovsky by means of model construction, are mechanisms, since they all represent only particular variants of the indicated six-link spatial mechanism with an axis of symmetry.

1. “Proceedings of the Tomsk Polytechnic Institute,” vol. 61, issue 1, 1947. ↩