Mechanisms are prone to stuttering or even jamming due to the deformation of components, especially for overconstrained linkages for which geometric conditions must be strictly satisfied. In this paper, joint clearance is actively introduced to release the local constraint so that the linkages can still achieve general movement under the deformation of components. And a joint clearance model with a revolute joint (

Overconstrained linkages, which do not satisfy the Grübler–Kutzbach (G-K) criterion, are a kind of special mechanism and still have mobility due to the special geometric relationship between the joints and links. They have great application potential, especially in compact-deployable mechanisms, for their excellent rigidity and folding performance.

Over the past few decades, there have been many overconstrained linkages found. In Sarrus (1853), the first overconstrained linkage, the Sarrus linkage, was found. Since then, Bennett (1903), Bricard (1927), Myard (1931), and Goldberg (1943) linkages have been proposed – one after another. Based on those overconstrained linkages proposed, scholars have done a lot of research. Baker (1979) analyzed the intrinsic relationship between Bennett (1903), Goldberg (1943), and Myard (1931) and derived the complete closure equation for each linkage. In Song et al. (2014), Feng et al. (2017), and Yang and Chen (2021), the kinematic equation of the Bricard (1927) linkage was derived, and the bifurcation of motion under different geometric conditions was analyzed.

However, the strict geometric conditions of overconstrained linkages limit
their practical application in spatial mechanisms (Mavroidis and Beddows, 1997; Chen, 2003), especially for the deployable structures constructed with overconstrained linkages that are expected to work in a space with a large change in temperature. To overcome this disadvantage, Yang et al. (2016) proposed a general truss transformation method to find their non-overconstrained forms and proved that the non-overconstrained equivalent linkages of the Bennett (1903) linkage and
Myard (1931) 5

In this paper, joint clearance is actively introduced to make overconstrained linkages movable, even when their components have deformations. For convenience, only one joint is allowed to introduce clearance, and the target is to find the minimum clearance that avoids too much wiggling. A brief description of the joint clearance model employed by the proposed method is first presented in Sect. 2, including an analysis of the relative motion between its journal and bearing. Section 3 presents the solving process of the minimum clearance in detail. In Sect. 4, a case study of the Bennett linkage is presented. And conclusions are given in Sect. 5 end the paper.

A revolute joint (hereafter

Spatial

There are, in total, three types of contact states between the journal and the bearing, namely surface contact, line contact, and point contact.

When there is only transition for the bearing along its axis, then the end plane of the bearing will get into touch with the end face of the journal and its the surface contact state (see Fig. 2d).

When there is only motion along the radial direction to the left, then it is named the line contact state. The bearing wall is fitted to the cylindrical surface of the journal (as shown in Fig. 2c).

The third contact state will appear when there is a tilt angle between the axial and the radial axes. The cylindrical surface of the journal may be in contact with the bearing wall (as shown in Fig. 2a), and the upper (or lower) end face of the journal may get in contact with the end plane of the bearing (as shown in Fig. 2b), which is called the point contact.

Contact states, including

In addition, a combination of line contact and surface contact may occur.

In terms of large deployable mechanisms, they usually move at a low speed, which means that the journal and bearing are stably contacted during the whole motion. That is to say, there are at least two points that are in contact to guarantee stability. Significantly, coming into contact on the upper side or the lower side is the same due to the symmetry. Similarly, the bearing's tilting to the left or the right can be also viewed as being the same case.

Therefore, the contacting modes can be represented by one point indicating the position and posture of the axis of the bearing. For instance, the upper-right point of every subgraph in Fig. 2 is selected to indicate the contact between the journal and the bearing.

In such a manner, the contact modes can be divided into two cases, C

Contact modes, including

In Fig. 4,

Coordinate systems.

There is a tendency to move in five other directions when there are radial and axial clearances in an

Relative movement states, including the

Initially, the point

where

The bearing has

Incline plane schematic, with the

From the description above, the point can be obtained as follows:

The transformation matrix for the rotating the angle

In

From Fig. 6b, the following can be obtained:

When the contact mode is C

The continuous movable condition of a linkage is that each input has a corresponding output. The geometric conditions of overconstrained linkages are not satisfied any more when the components have a deformation, which would cause linkage to become stuck or even to become rigid. Proper joint clearance will remove the need for the geometric conditions and make it possible for the linkages to move.

The larger the value of the clearance introduced, the higher tolerance of the range of deformation for the components can be – in theory. However, as clearance would cause fluctuations and reduce motion accuracy, it cannot be enlarged too much. So, the target is to find the minimum value of a joint clearance below which the mechanism could not work for the whole working period; i.e., the linkages' critical condition is that there is only one set of inputs and outputs satisfying a one-to-one correspondence at some particular position.

Linkages are separated into the

Due to the property of a single degree of freedom, single-loop, overconstrained linkages are divided into two parts, namely input and output ones, based on the location introduced for joint clearance, as shown in Fig. 7. The minimum clearance-introducing value can be obtained by judging whether the linkages meet continuous movable and critical conditions. Besides, the location of the clearance introducing does not include the input and output joints. And the bisection method is used to adjust the clearance in order to approach it constantly. The flowchart is shown in Fig. 8, and the specific steps are as follows.

The flowchart of minimum clearance.

The precision and efficiency of the numerical iterative method are affected by the step distance of the variable. So, two steps are adopted to find the final results, namely preliminary search and final search, where the preliminary search has a larger step distance, and the final search has a smaller one.

In this section, the Bennett linkage is used as an example to analyze the minimum clearance introduced when it has the deformation of certain components. The Bennett linkage is a typical spatially overconstrained mechanism with the least number of links (Yang et al., 2020; as shown in Fig. 9), and its geometric condition without joint clearances is as follows (Bennett, 1903):

Bennett linkage.

This paper takes the commonly used space material, titanium alloy (TC4), as
an example for analysis. And the geometric parameters and the components'
deformation of the Bennett linkage are listed in Table 1, where the
components' deformation is obtained by a liner thermal expansion formula and
Hooke's law (Hongwen, 2017). Here, the limit of the temperature difference in
the space

Related parameters of the Bennett linkage.

According to the analysis, it requires at least 0.379 mm clearance at joint B to keep the Bennett linkage's motion smooth, and it is the same at joint C. So, introducing clearance at joint B or joint C is equivalent in the Bennett linkage. Figure 10 shows the fluctuation in the output angle after the introduction of clearance in joint B and the components' deformation. It shows a trend of high clearance on both sides, it is low in the middle in each half-motion cycle, and the fluctuation is close to 0 when the input is around

The fluctuation in the output angle.

The fluctuation tends to 0, which indicates that the position around 109

Relationship of the workspace between input and output parts when

This means that the current input and output have a corresponding relationship when the two workspaces intersect. And it can be seen from Fig. 11a that the two workspaces are fairly close, and the minimum distance is 0.0191 mm, which can be regarded as an intersection. However, the minimum distance between the two workspaces is 20.2552 mm (Fig. 11b), which cannot be ignored. So, the current input corresponds to a small range of output; i.e., the introduced clearance allows the Bennett linkage to meet the critical condition at the present position. Therefore, the result is the minimum clearance introduced under the current deformation of the components.

In this paper, we present a joint clearance model with an

Taking the Bennett linkage as an example, the optimal position to introduce clearance is analyzed, and the result shows that the clearance introduced at joint B is the same as joint C under the same deformation. That is, both joint B and joint C can meet the critical conditions of the Bennett linkage with a smaller joint clearance, so that the Bennett linkage can achieve the alleviation of a freezing problem under smaller fluctuations in the output. However, it may be different in other overconstrained linkages. In future work, it is necessary to study the relationship between the size of the clearance introduced and the deformation of the component, so as to apply the method more effectively.

The data are available upon request from the corresponding author.

FY is the lead author of this paper. FY and JQ wrote the draft. JQ completed the joint modeling and the method analysis verification under the supervision of FY. YG checked and revised the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Fufu Yang acknowledges the financial support from the National Natural Science Foundation of China (grant no. 51905101), Fuzhou University Testing Fund of Precious Apparatus (grant no. 2022T014), and the Natural Science Foundation of Fujian Province, China (grant no. 2019J01209).

This research has been supported by the National Natural Science Foundation of China (grant no. 51905101), the Natural Science Foundation of Fujian Province (grant no. 2019J01209), and Fuzhou University Testing Fund of Precious Apparatus (grant no. 2022T014).

This paper was edited by Chin-Hsing Kuo and reviewed by two anonymous referees.