This contribution shows that a method proposed previously, for the determination of the instantaneous centers of rotation of planar closed chains, can be generalized for the determination of the instantaneous screw axes of general one-degree-of-freedom spatial mechanisms. Hence, the approach presented in this paper can be applied to any of the closed chains that belong to any of the subgroups of the Euclidean group, SE(3), namely planar, spherical or chains associated with the Schönflies subgroups, among others. Furthermore it can be also applied to multi-loop mechanisms and even to closed chains that are exceptional o paradoxical, as indicated by Hervé.

In 2016,

The instantaneous centers of velocity or rotation are defined as a pair of coincident points that belong to different links of the mechanism such that one link rotates with respect to the other around an axis perpendicular to the plane of motion and passes through the pair of coincident points. Instantaneous centers that can be determined by the application of the very same definition are denoted as

Since the first decades of the XX century, it was well known, see

It is important to note that despite the references cited by

In the last paragraph of the second column of page 1,

Here,

In the fourth paragraph of the first column of page 2,

In several parts,

In 1992,

In the rest of this paper the method presented by

In this section, the fundamentals of the velocity analysis of single-loop and multi-loop linkages, using screw theory, will be briefly reviewed, see

The velocity state of a rigid body

It can be proved that for

Consider the single-loop closed chain shown in Fig.

Single-loop closed chain.

If link

If the mechanism is multi-loop, it is necessary to resort to some basics of graph theory, see

Directed graph of the single flyer planar mechanism.

Single flyer planar mechanism.

The multi-loop mechanism has three independent loops. However, there are other selections besides those shown in Fig.

This is the equation used in the velocity analysis of the indeterminate planar mechanism dealt with in Sect.

This section illustrates the computation of the instantaneous screw axes, or their corresponding simplifications, of three representative linkages. In all these examples, the velocity analysis, for an arbitrary velocity of the driver link will be solved, and from the results, the velocity state of an arbitrary link

Consider the mechanism shown in Fig.

Instantaneous centers of velocity associated with the single flyer linkage.

In order to carry out the velocity analysis, it is necessary to determine the screws associated with the kinematic pairs, all of them revolute pairs, with respect to the origin of the coordinate system,

In this contribution all vectors are column vectors; however due to space considerations they are written as the transpose of row vectors.

, theThe screws associated with the kinematic pairs, with respect to the origin,

Once the velocity analysis has been completed the velocity states between two arbitrary links can be computed and the corresponding instantaneous screw axes can be determined. Due to space considerations, only a few instantaneous velocity centers will be computed; for example:

First Case. Instantaneous center of velocity

Second Case. Instantaneous center of velocity

The locations of the instantaneous velocity centers

Consider the multi-loop spherical mechanism proposed by

Instantaneous axes of velocity associated with the indeterminate spherical mechanism.

It should be noted that, unlike the planar mechanism, here the subscripts associated with the primary instantaneous axes of velocity are chosen in the way that makes simpler to understand the required velocity analysis equations.

Indeterminate spherical mechanism.

Figure

Directed graph of the single flyer spherical mechanism.

Assuming that

From these results it is straightforward to compute the velocity state between two arbitrary links of the spherical mechanisms and eventually determine the corresponding instantaneous axis of velocity. For example:

First Case. Instantaneous axis of velocity

Second Case. Instantaneous axis of velocity

Consider the RCCC spatial mechanism shown in Fig.

The RCCC spatial mechanism.

The velocity analysis equation is given by

From these results, it is possible to determine the velocity states of the different links relative to the other links, and from these velocity states the corresponding instantaneous screw axes, ISA will be determined. One special and simplest case is the ISA of the relative movement of link

Now, the the ISA of the relative movement of link

This paper has shown that once the velocity analysis of an arbitrary mechanism with one degree of freedom has been carried out it is straightforward to determine the instantaneous screw axis associated with the relative movement of two arbitrary links of the mechanism, regardless the type of mechanism, whether they are single- or multi-loop, or they are determined or undetermined. However, it should be noted that in kinematics, the usual approach is to find first the instantaneous screw axis associated with two arbitrary links of the mechanism and from this knowledge to perform the mechanism's velocity analysis. An upcoming paper will analyze thoroughly this approach for arbitrary mechanisms. All the computations were carried out using Maple© and verified with Adams©.

The complete thesis of Juan Ignacio Valderrama-Rodríguez is
located in

The present work originated from the Master thesis of JIVR. JMR and JJCS were the co-directors of the thesis. It can be assured that the authors worked proportionally to the thesis and the present paper.

The authors declare that they have no conflict of interest.

Juan Ignacio Valderrama-Rodríguez thanks the Conacyt, the Mexican National Council of Science and Technology, for the support through a grant no. 458523 to pursue a MSc degree at the Universidad of Guanajuato. The authors thank Conacyt and the Universidad of Guanajuato, through DICIS and the Mechanical Engineering Department, for their continuous support.

This research has been supported by the Mexican National Council of Science and Technology (grant no. 458523).

This paper was edited by Daniel Condurache and reviewed by Soheil Zarkandi and two anonymous referees.