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  • Original Article
  • Open Access

An Error Equivalent Model of Revolute Joints with Clearances for Antenna Pointing Mechanisms

Chinese Journal of Mechanical Engineering201831:39

https://doi.org/10.1186/s10033-018-0233-6

  • Received: 29 June 2017
  • Accepted: 16 April 2018
  • Published:

Abstract

Joint clearances in antenna pointing mechanisms lead to uncertainty in function deviation. Current studies mainly focus on radial clearance of revolute joints, while axial clearance has rarely been taken into consideration. In fact, owning to errors from machining and assembly, thermal deformation and so forth, practically, axial clearance is inevitable in the joint. In this study, an error equivalent model (EEM) of revolute joints is proposed with considering both radial and axial clearances. Compared to the planar model of revolute joints only considering radial clearance, the journal motion inside the bearing is more abundant and matches the reality better in the EEM. The model is also extended for analyzing the error distribution of a spatial dual-axis (“X–Y” type) antenna pointing mechanism of Spot-beam antennas which especially demand a high pointing accuracy. Three case studies are performed which illustrates the internal relation between radial clearance and axial clearance. It is found that when the axial clearance is big enough, the physical journal can freely realize both translational motion and rotational motion. While if the axial clearance is limited, the motion of the physical journal will be restricted. Analysis results indicate that the consideration of both radial and axial clearances in the revolute joint describes the journal motion inside the bearing more precise. To further validate the proposed model, a model of the EEM is designed and fabricated. Some suggestions on the design of revolute joints are also provided.

Keywords

  • Error modeling
  • Joint clearances
  • Antenna pointing mechanism
  • Radial clearance
  • Axial clearance

1 Introduction

In order to achieve real-time tracking and precise pointing on target satellites, dual-axis antenna pointing mechanisms have been widely applied in communication satellites and data relay satellites for satellite-ground and satellite-satellite communication and data transmission. Pointing accuracy of antenna pointing mechanisms plays an important role in dictating the efficiency of the satellite communication system. Since the pointing accuracy is affected by composite factors including joint clearances, thermal load, etc., achieving a high pointing accuracy is a challenging task especially in space.

Clearances in mechanical joints which inevitably exist in all kinks of machines for instance dual-axis antenna pointing mechanisms, on one hand, significantly influence performance of the mechanism [1, 2]; on the other hand, they are indispensable to allow relative motion between parts and to enable component assemblage.

A large number of researchers have studied on the subject associated with clearance joints [36]. Three main methods dealing with clearance in revolute joints are proposed, the spring-damper approach, the massless link approach and the contact force approach [7]. Wang et al. [8] presented a method to determinate panel adjustment values from far field pattern in order to improve the accuracy of large reflector antenna. You et al. [9] modeled and analyzed satellite antenna systems considering the influences of joint clearances and reflector flexibility. Deducing from the joint clearance of manipulators, Ting et al. [10] proposed a simple approach to identify the largest position and direction errors. Zhang et al. [11] provided a dynamic model with multiple clearances of planetary gear joint to analyze the vibration characteristics. By using “contact-separation” two-state model, Li et al. [12] established the multibody system dynamic equations of two-dimensional pointing mechanism with clearance. Zhang et al. [13] studied the comparison on kinematics and dynamics between the fully actuated 3-RRR mechanism and the redundantly actuated 4-RRR mechanism with joint clearances. According to the probability theory, Zhu et al. [14] presented the uncertainty analysis of robots with revolute joint clearances, which can be applied in both planar and spatial mechanical systems. Bai et al. [15] established a hybrid contact force model to forecast the dynamic performance of planar mechanical systems with revolute joint clearances. Venanzi and Parenti-Castelli [16] proposed a method to evaluate the influence of clearances on accuracy of mechanisms, which works for both planar and spatial mechanisms. Within the framework of finite element, Bauchau et al. [17, 18] presented a method to model planar and spatial joints with clearances. Brutti et al. [19] presented a general computer-aided model of a 3D revolute joint with clearances suitable for implementation in multi body dynamic solvers. Taking both radial and axial clearances into consideration, Yan et al. [20] established a synthetic model for 3D revolute joints with clearances in mechanical systems by the contact force approach. Based on the contact force approach, Marques et al. [21, 22] presented a formulation to model spatial revolute joints with radial and axial clearances. In the past decades, modelling of revolute joints with clearances has attracted a wide investigation since it is a significant factor in prediction of kinematic and dynamic performance of mechanical systems. However, most of them only focused on planar revolute joints [2329], which means only radial clearance has been considered, axial clearance has been scarcely taken into discussion. In fact, because of errors from machining and assembly, thermal deformation and so forth, axial clearance also occurs inevitably in the joint which could cause out-of-plane motion between the journal and the bearing. Combination of both radial and axial clearances in the revolute joint would make the journal motion inside the bearing more complex and unpredictable.

In this paper, by assuming the radial clearance as a virtual massless link with variable length, an error equivalent model (EEM) of revolute joints with clearances is presented, in which both radial and axial clearances are taken into consideration. Compared to the planar model of revolute joints only with radial clearance, journal motion inside the bearing is more abundant and matches the reality better in this model. Besides, the model is more intuitive and graphic than the 3D contact force model of revolute joints in Refs. [2022] and it is easier to be applied to analysis of pointing accuracy of the spatial dual-axis pointing mechanism (Additional file 1).

2 Modeling of Revolute Joints with Clearances

2.1 Revolute Joints with Clearances

In theoretical mechanical systems, it is assumed that the rotational axes of two bodies, which are linked by an ideal revolute joint, are coaxial. In practice, there is an unavoidable clearance between the journal and bearing for allowing rotation. Taking the joint clearance into consideration, a relative motion between the two axes, under the constraint of the bearing boundary, will be generated. Figure 1 depicts a cross-section of a revolute joint with radial clearance and its equivalent model [30, 31]. In the cross-section, the difference between radii of the bearing and the journal defines the range of radial clearance.
Figure 1
Figure 1

Revolute joints with radial clearance

In Figure 1(a), the radial clearance can be described as k,
$$\left\{ \begin{aligned} &0 \le k \le k_{\text{max} } , \hfill \\ &k_{\text{max} } = R_{\text{b}} - R_{\text{j}} , \hfill \\ \end{aligned} \right.$$
(1)
where \(R_{\text{b}}\) and \(R_{\text{j}}\) are radii of the bearing and the journal, respectively.

Figure 1(b) depicts an equivalent model of the revolute joint with radial clearance, on a cross section, by treating the radial clearance as a virtual massless link with variable length. Relative to the situation of an ideal joint, the joint clearances introduce two extra degrees of freedom in the mechanical system, which can be described by a combination of a translation and a rotation. Movement between the journal and the bearing in the range of the radial clearance can be treated as that of a RPR mechanism. In Figure 1(b), O and O1 indicate the center of the bearing and journal, respectively. Joints A and C are ideal revolute joints and joint B is a prismatic joint. Range of motion of the prismatic joint is limited by the maximum radial clearance, that is [0, kmax]. θ is the angle from the x axis to the line between the center of bearing and the center of journal, \(l_{{OO_{1} }}\), θ[0, 2π]. Therefore, in O-xy, coordinate of the journal center is (kcosθ, ksinθ).

In previous studies, most researchers described the revolute joint with clearances as planar mechanisms. The journal can perform only translational motion with respect to the bearing. However, it is obviously that the assumption does not accord with the physical truth. When joint clearances exist, apart from translation, relative rotation between the bearing and journal can also appear. Therefore, placement of the cross-sections of the journal on two end faces of the bearing can be different which generates different positions and orientations of the journal.

Comparing with the length of a revolute joint, the maximum radial clearance is relatively small. Although, on the end face, projection of cross-section of the journal varies from circle to ellipse along with the rotation between the journal and the bearing, the variation is small enough to be ignored. In the following, we assume that the tiny change does not affect the range of radial clearance, which means that the cross-section of the journal on each end face remains circle. Possible distributions of the journal are shown in Figure 2.
Figure 2
Figure 2

Possible distributions of the journal

Figure 2(a) describes the journal only performs translational motion under the constraint of the bearing boundary. Figures 2(b) and 2(c) show the journal has both the translational motion and rotational motion while rotational axis of the journal is still on or parallel to the O-xz plane or O-yz plane. General distribution of the journal is illustrated in Figure 2(d).

2.2 EEM of the Revolute Joint with Clearances

Known from Section 2.1, physical motion of the journal is relatively complicated. The planar model is inappropriate for obtaining the real error. To more precisely analyze the revolute joint with radial and axial clearances, an EEM of the revolute joint with clearances is established, as shown in Figure 3.
Figure 3
Figure 3

An EEM of the revolute joint with clearances

Movement of the journal due to radial and axial clearances is described as that of the platform of a 2RPU-C mechanism. In Figure 3, O1 and O2, which are the center of the two revolute joints respectively, are fixed. Distance between O1 and O2 is l. O3 and O4 are defined as center of the two U hinges, respectively. O5 is the midpoint of segment \(l_{{O_{3} O_{4} }}\). The reference plane is parallel to the base plane of the revolute joint and passes O1. Plane 1 is parallel to the reference plane and passes O4. O’3 indicates the projection of O3 on Plane 1. \(l_{{O^{\prime}_{3} O^{\prime}_{4} }}\) and \(l_{{O^{\prime}_{4} O_{4} }}\) are on Plane 1 and perpendicular with each other. Plane 2 is determined by O3, O4 and O’4. It can also be obtained by rotating Plane 1 around \(l_{{O^{\prime}_{4} O_{4} }}\) with an angle − α, as shown in Figure 3. O1-x1y1z1 is the reference coordinate system. z1 axis is along the ideal rotational axis of the revolute joint, x1 axis is perpendicular to the reference plane. The O2-x2y2z2 coordinate system is obtained by translating the O1-x1y1z1 along z1 axis with a distance, − l, then rotating an angle, θ2, around z1 axis. x5 axis is perpendicular to Plane 2 and z5 axis is along the direction of \(l_{{O_{3} O_{4} }}\). In Figure 3, k i (i=1, 2) is defined as the translational distance of the ith prismatic joint, k i [0, kimax]. θ1 is the angle from x1 axis to the direction of the first prismatic joint, θ1,θ2[0, 2π]. In the EEM, the line between O1 and O2 (z1 axis) indicates the ideal axis of the revolute joint, while the physical axis is along the line between O3 and O4 (z5 axis). When k1=0, k2=0, the ideal rotational axis and the physical axis are coincident. Known from kinematics of the 2RPU-C mechanism, the physical journal performs both the translational motion and rotational motion. To further validate the proposed model, a model of the EEM is designed and fabricated. Prototype of the EEM is presented as shown in Figure 4.
Figure 4
Figure 4

Prototype of the EEM

Configurations of the prototype shown in Figure 4 are corresponding to those in Figure 2. In Figure 4(a), the physical journal has only translational motion. Figures 4(b) and 4(c) show configurations that the journal acts both translational and rotational motion while physical journal of the revolute joint is parallel to the reference plane or the O-x1z1 plane constantly. A general configuration of the physical journal is depicted in Figure 4(d). Comparing with the planar clearance model of revolute joints, the EEM of the revolute joint with clearances is more precise since it contains the relative rotation between journal and bearing.

3 Analysis of the EEM of Revolute Joints with Clearances

3.1 Homogeneous Transformation Matrix of the EEM

Section 2 has presented an EEM of the revolute joint with clearances. As shown in Figure 3, if we define that \(O_{i}^{j} \, = \, [x_{{O_{i} }} ,\,y_{{O_{i} }} ,z_{{O_{i} }} ]^\text{T}\) is the coordinate of O i in O j -x j y j z j . Direction of the ideal journal is assumed to be [0, 0, 1]T in O1-x1y1z1. It is obvious that,
$$\left\{ \begin{aligned} &O_{3}^{2} { = [}k_{2} ,\;0,\;0]^\text{T} , \hfill \\ &O_{4}^{1} { = [}k_{1} \cos \theta_{1} ,\;k_{1} \sin \theta_{1} ,\;0]^\text{T} . \hfill \\ \end{aligned} \right.$$
(2)
Coordinate of O3 in O1-x1y1z1 can be calculated as
$$\left( {\begin{array}{*{20}c} {O_{3}^{1} } \\ 1 \\ \end{array} } \right){ = }\left( {\begin{array}{*{20}c} {\cos \theta_{2} } & { - \sin \theta_{2} } & 0 & 0 \\ {\sin \theta_{2} } & {\cos \theta_{2} } & 0 & 0 \\ 0 & 0 & 1 & { - l} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k_{2} } \\ 0 \\ 0 \\ 1 \\ \end{array} } \right).$$
(3)

Therefore, \(O_{3}^{1}\) = [k2cosθ2, k2sinθ2, − l]T.

In O1-x1y1z1, direction of the physical journal, n, can be described by the vector from O3 to O4, which is
$$\varvec{n}{ = }O_{4}^{1} - O_{3}^{1} { = }\left( {\begin{array}{*{20}c} {k_{1} \cos \theta_{1} - k_{2} \cos \theta_{2} } \\ {k_{1} \sin \theta_{1} - k_{2} \sin \theta_{2} } \\ l \\ \end{array} } \right)^{\text{T}} .$$
(4)

As shown in Figure 3, x5 axis is perpendicular to Plane 2 and z5 axis is along the direction of \(l_{{O_{3} O_{4} }}\).

It is known that the vector from O3 to O4 in O1-x1y1z1 is n. Similarly, the normal vector of Plane 2 in O1-x1y1z1 is
$$\varvec{p} = (\cos \alpha ,0, \sin \alpha ).$$
(5)
Then direction of y5 axis can be calculated as
$$\varvec{q} = \varvec{n} \times \varvec{p} .$$
(6)
As shown in Figure 3, we can gain the following equations,
$$\left\{ \begin{gathered} O{^{\prime}_{3}}^{1} = {\text{[}}k_{1} \cos \theta _{1} {\text{,}}\;k_{2} \sin \theta _{2} ,\; - l]^\text{T} , \hfill \\ O{^{\prime}_{4}}^{1} = {\text{[}}k_{1} \cos \theta _{1} {\text{,}}\;k_{2} \sin \theta _{2} ,\;0]^\text{T} , \hfill \\ \end{gathered} \right.$$
(7)
$$O_{5}^{1} { = }\left[ {\begin{array}{*{20}c} { (k_{1} \cos \theta_{1} + k_{2} \cos \theta_{2} )/2} \\ { (k_{1} \sin \theta_{1} + k_{2} \sin \theta_{2} )/2} \\ { - l/2} \\ \end{array} } \right].$$
(8)
Meanwhile, O5-x5y5z5 can be constructed by the following three steps:
Step 1:: 

Translate the O1-x1y1z1 coordinate system to O5;

Step 2:: 

Rotate the coordinate system produced in Step 1 with an angle, − α, around its y axis;

Step 3:: 

Rotate the coordinate system produced in Step 2 with an angle, β, around its x axis

Then we can get
$$_{1}^{5} T{ = }TR_{y} (- \alpha )R_{x} (\beta ),$$
(9)
where \(_{1}^{5} T\) defines the homogeneous transformation matrix from O5-x5y5z5 to O1-x1y1z1,
$$\tan \alpha = l_{{O_{3} O^{\prime}_{3} }} /l_{{O^{\prime}_{3} O^{\prime}_{4} }} = (k_{2} \cos \theta_{2} - k_{1} \cos \theta_{1} )/l,$$
$$\tan \beta = l_{{O_{4} O^{\prime}_{4} }} /l_{{O_{3} O^{\prime}_{4} }} \approx l_{{O_{4} O^{\prime}_{4} }} /l_{{O^{\prime}_{3} O^{\prime}_{4} }} = (k_{2} \sin \theta_{2} - k_{1} \sin \theta_{1} )/l,$$
$$\alpha = \text{arctan}\left[ {(k_{2} \cos \theta_{2} - k_{1} \cos \theta_{1} )/l} \right],$$
$$\beta = \arctan \left[ {(k_{2} \sin \theta_{2} - k_{1} \sin \theta_{1} )/l} \right].$$
$$R_{y} (- \alpha ) { = }\left( {\begin{array}{*{20}c} {\cos \alpha } & 0 & { - \sin \alpha } & 0 \\ 0 & 1 & 0 & 0 \\ {\sin \alpha } & 0 & {\cos \alpha } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$
(10)
$$R_{x} (\beta ) = \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & {\cos \beta } & { - \sin \beta } & 0 \\ 0 & {\sin \beta } & {\cos \beta } & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$
(11)
$$T{ = }\left( {\begin{array}{*{20}c} 1 & 0 & 0 & { (k_{1} \cos \theta_{1} + k_{2} \cos \theta_{2} )/2} \\ 0 & 1 & 0 & { (k_{1} \sin \theta_{1} + k_{2} \sin \theta_{2} )/2} \\ 0 & 0 & 1 & { - l/2} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right).$$
(12)
Therefore,
$${}_{1}^{5} T{ = }\left( {\begin{array}{*{20}l} {\text{c} \alpha }&{- \text{s} \alpha \text{s} \beta }&{ - \text{s} \alpha \text{c} \beta }&{a} \\ 0&{\text{c} \beta }&{- \text{s} \beta }&{b} \\ {\text{s} \alpha }&{\text{c} \alpha \text{s} \beta }&{ \text{c} \alpha \text{c} \beta }&{- l/2} \\ 0&0& 0&{1} \\ \end{array} } \right),$$
(13)
where \(a = (k_{1} \cos \theta_{1} + k_{2} \cos \theta_{2} )/2,\)\(b = (k_{1} \sin \theta_{1} + k_{2} \sin \theta_{2} )/2,\)\({\text{c}}\alpha = { \cos }\alpha , {\text{ s}}\alpha = \sin \alpha ,\;\;{\text{c}}\beta = { \cos }\beta , {\text{ s}}\beta = \sin \beta.\)

3.2 Constraint Analysis

In the EEM, the revolute joint with clearances is equivalent to a 2RPU-C mechanism. The two revolute joints on the base plane are assumed to be fixed. Distance between O3 and O4 is defined as \(l_{{O_{3} O_{4} }} { = }l + \Delta l\). l is the distance between O1 and O2, as well as the distance between O3 and O4 along with z1 axis. \(\Delta l\) is defined as the comprehensive axial clearance, such as axial clearance of machining and axial thermal deformation, etc. Distance between O3 and O4 can be also calculated as,
$$l^{\prime}_{{O_{3} {\text{O}}_{4} }} = \sqrt {k_{1}^{2} + k_{2}^{2} - 2k_{1} k_{2} \cos (\theta_{1} - \theta_{2} )+ l^{2} }.$$
(14)
When \(\theta_{2} { = }\theta_{1} { + }\uppi\), the distance reaches its maximal value,
$$l^{\prime}_{{O_{3} O_{4} \text{max} }} = \sqrt {\left(k_{{1{ \text{max} }}} + k_{{2{ \text{max} }}} \right)^{2} + l^{2} }.$$
(15)
Herein, we define \(\Delta l^{\prime}\) as,
$$\Delta l^{\prime}{ = }l^{\prime}_{{O_{3} O_{4} \text{max} }} - l.$$
(16)

When \(\Delta l\, \ge \,\Delta l^{\prime}\), the physical journal can freely carry out the translational and rotational motion constrained within the boundary of bearing. However, if \(\Delta l\, < \,\Delta l^{\prime}\), movement of the physical journal will be limited.

4 EEM of the Dual-axis (“XY” Type) Antenna Pointing Mechanism with Revolute Joint Clearances

Normally, the dual axes of the antenna pointing mechanism, such as the “X–Y” type antenna pointing mechanism, are orthogonal to each other. Based on the EEM of the revolute joint discussed in Section 2, an EEM of the “X–Y” type antenna pointing mechanism with revolute joint clearances is proposed in this section.

The EEM of the “X–Y” type antenna pointing mechanism can be treated as a combination of two error equivalent models of single revolute joint linked through a prismatic joint, as shown in Figure 5. The ideal journal of the revolute joint Y is orthogonal to the physical journal of the revolute joint X.
Figure 5
Figure 5

An EEM of the “X–Y” type antenna pointing mechanism with revolute joint clearances

In Figure 5(a), O11 is the midpoint of segment \(l_{{O_{9} O_{10} }}\); O7 and O8 express the center of revolute joints in the revolute joint Y, respectively. O6 is the midpoint of segment \(l_{{O_{7} O_{8} }}\). Meanwhile, O9 and O10 are defined as the center of the U type hinges in the equivalent model of joint Y. Plane 3 is parallel to Plane 2, it also passes O7. In addition, \(l_{{O_{5} O_{6} }}\)is perpendicular to both Plane 2 and Plane 3. \(l_{{O_{7} O_{8} }}\) and \(l_{{O_{3} O_{4} }}\) are orthogonal to each other.

Similarly, we can get the homogeneous transformation matrix from O11-x11y11z11 to O7-x7y7z7,
$$_{7}^{{11}} T = \left( {\begin{array}{*{20}c} {{\text{c}}\omega } & { - {\text{s}}\omega {\text{s}}\tau } & { - {\text{s}}\omega {\text{c}}\tau } & m \\ 0 & {{\text{c}}\tau } & { - {\text{s}}\tau } & n \\ {{\text{s}}\omega } & {{\text{c}}\omega {\text{s}}\tau } & {{\text{c}}\omega {\text{c}}\tau } & { - l/2} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$
(17)
where \(\omega = \text{arctan}\left[ {(k_{8} \cos \theta_{8} - k_{7} \cos \theta_{7} )/l} \right],\)\(\tau = \arctan \left[ {(k_{8} \sin \theta_{8} - k_{7} \sin \theta_{7} )/l} \right],\)
$$m = (k_{7} \cos \theta_{7} + k_{8} \cos \theta_{8} )/2,$$
$$n = (k_{7} \sin \theta_{7} + k_{8} \sin \theta_{8} )/2.$$
The coordinate system O7-x7y7z7 can be obtained through the following three steps:
Step 1:: 

Translate O5-x5y5z5 with a distance, k5, along x5 axis.

Step 2:: 

Translate the coordinate system obtained in Step 1 with a distance, l/2, along y5 axis.

Step 3:: 

Rotate the coordinate system obtained in Step 2 with an angle, −  π/2, around its x axis.

Therefore,
$$O_{7}^{5} = \left[ {k_{5} ,l/2,0} \right]{}^{\text{T}}.$$
(18)
Homogeneous transformation matrix from O7-x7y7z7 to O5-x5y5z5 is:
$${}_{5}^{7} T{ = }\left( {\begin{array}{*{20}c} 1 & 0 & 0 & {k_{5} } \\ 0 & 0 & 1 & {l/2} \\ 0 & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right),$$
(19)
where k5 is the distance between \(l_{{O_{7} O_{8} }}\) and \(l_{{O_{3} O_{4} }}\). It is mainly affected by the thermal load.
From the above derivation, homogeneous transformation matrix from O11-x11y11z11 to O1-x1y1z1 can be obtained
$${}_{1}^{11} T = {}_{1}^{5} T{}_{5}^{7} T{}_{7}^{11} T.$$
(20)

5 Case Studies

In this section, several case studies are performed based on the proposed EEM. Different values of the geometric parameters are adopted as given in Table 1.
Table 1

Parameters of the EEM in case studies

Parameter

Case 1

Case 2

Case 3

\(\Delta l\) (mm)

0

0.01

0.1

k1 (mm)

[0, 0.05]

[0, 1]

[0, 1]

k2 (mm)

[0, 0.05]

[0, 1]

[0, 1]

θ1 (rad)

[0, 2π]

[0, 2π]

[0, 2π]

θ2 (rad)

[0, 2π]

[0, 2π]

[0, 2π]

l (mm)

180

180

180

5.1 Case 1

In the first case, \(\Delta\) l= 0 mm, it means that the revolute joint has only radial clearance and no axial clearance. \(\Delta l=0\,\text{mm}\) leads to k1 = k2 all along. Therefore, the journal can perform only translational motion which indicates the EEM of the revolute joint with clearances degenerating into the traditional planar model. The ideal journal axis and the physical journal axis are always parallel. Figure 6 depicts the configuration of the journal only have radial clearance.
Figure 6
Figure 6

A possible configuration of the journal only with radial clearance

Since axes of the ideal journal and the physical journal are always parallel to each other, if we define the distance between the two axes is d, it is obviously that dmax = k1max = 0.05 mm.

5.2 Case 2

\(\Delta\)l = 0.01 mm in the second case. From Eqs. (15) and (16), by using the parameters in Table 1, the following can be obtained,
$$l^{\prime}_{{O_{3} O_{4}{_\text{max} }}} = 180.011\;{\text{mm,}}$$
$$\Delta l^{\prime}{ = }l^{\prime}_{{O_{3} O_{4} {_{\text{max}} }}} - l = 0.011\;{\text{mm}} .$$

It is obvious that \(\Delta l\, < \,\Delta l^{\prime}\), which limits the movement of the physical journal. The radial clearance is restricted by the axial clearance which results in the restrains of k i and θ i . In the 2RPU-C mechanism, the two branches are equivalent. It is allowable to discuss only one of them. Clearances in the RPU branch can be divided into the following two sub-cases.

Sub-case 1: k1 keeps its range from 0 to 1 mm, which allows it to reach its maximum value. θ1 and θ2 still range from 0 to 2π rad. We set the values that k1 =k1max =1 mm and θ1 =π/2 rad, θ2 =θ1 +π=3π/2 rad. The sub-case is shown in Figure 7.
Figure 7
Figure 7

Distribution of the cross-section of the journal in sub-case 1

Herein, \(l_{{y - O_{3} O_{4} }}\) is the distance between O3 and O4 projected on O1-x1y1 plane:
$$l_{{y - O_{3} O_{4} }} = \sqrt {\left( {l + \Delta l} \right)^{2} - l^{2} } ,$$
(21)
$$k^{\prime}_{2\text{max} } = l_{{y - O_{3} O_{4} }} - k_{1\text{max} } .$$
(22)
Therefore, \(l_{{y - O_{3} O_{4} }} = 1.897\;{\text{mm}}\),
$$k^{\prime}_{2\text{max} } = 1.897 - 1{ = }0.897\;{\text{mm}} .$$

It can be seen that the maximum value of k2 reduces to 0.897 mm which indicates the k2 can only range from 0 to 0.897 mm.

Sub-case 2: we assume that both k1 and k2 can reach their maximum values (range from 0 to 1 mm). θ1 still ranges from 0 to 2π, while θ2 is restricted. Assuming that θ1 =π/2 rad, k1 =k2 =1 mm, configuration of the journal in sub-case 2 is shown in Figure 8.
Figure 8
Figure 8

Distribution of the cross-section of the journal in sub-case 2

Known from,
$$\varphi \,{ = }\,{ \arccos }\left[ { (k^{2}_{{ 1 {\text{max}}}} + k^{2}_{{ 2 {\text{max}}}} - l^{2}_{{y - O_{3} O_{4} }} )/2k_{{ 1 {\text{max}}}} k_{{ 2 {\text{max}}}} } \right].$$
(23)

It can be calculated that \(\varphi\)=0.795 rad, which means the angle between the line of the two prismatic pairs can only vary from 0 to 0.795 rad.

In case 2, if the inclination angle of the physical journal is defined as ψ, we can get
$$\psi{_{\text{max}} } = { \arctan }l_{{y - O_{3} O_{4} }} /l = 0.604^\circ ,$$
$$d{_{\text{max}} } = \,k_{\text{1max}} = 1\;{\text{mm}} .$$
For easier understanding, a possible configuration of the journal in case 2 is described in Figure 9.
Figure 9
Figure 9

A possible configuration of the journal in case 2

5.3 Case 3

In the third case, we have \(\Delta l\, \ge \,\Delta l^{\prime}\), which means the physical journal can freely realize both translational motion and rotational motion constrained within the bearing boundary. In general, the axial tolerance is large enough to allow the physical journal freely to perform rotational motion. Therefore, case 3 is more common in the practical engineering. Figure 10 displays a possible extreme position of the journal in case 3.
Figure 10
Figure 10

Configuration of the journal in case 3

The maximum distance and angle in case 3 is calculated as,
$$\psi{_{\text{max} }} = {\text{arctan}}\left[ { \, (k_{{\text{1max}}} + k_{\text{2max}} )/l} \right]{ = }0.637^\circ ,$$
$$d{_{\text{max}}} = k{_{\text{1max}}} = 1\;{\text{mm}} .$$
Through the above analysis, it can be known that when \(\Delta l\, = \,\Delta l^{\prime}\), the physical journal happens to freely realize both translational motion and rotational motion constrained within the bearing boundary. Figure 11 displays a possible extreme position of the journal when \(\Delta l\, = \,\Delta l^{\prime}\).
Figure 11
Figure 11

A possible configuration of journal when \(\Delta l\, = \,\Delta l^{\prime}\)

6 Conclusions

In this paper, an EEM of the revolute joint with clearances is proposed by the virtual bar method. First, the EEM of a single revolute joint with clearances, which is equivalent to a 2RPU-C mechanism, is established. Then, the model is extended for describing the error of a spatial dual-axis (“X–Y” type) antenna pointing mechanism. Comparing to the planar model of the revolute joint only with radial clearance, both radial clearance and axial clearance are involved. Due to the consideration of the rotational motion, the presented model can describe the error of revolute joints with clearances more precisely. It is also revealed that the radial and axial clearances are interrelated and restricted with each other. Three case studies on analyzing the internal relation between the radial and axial clearances are performed which puts forward some suggestions on the design of revolute joints.

Declarations

Authors’ Contributions

QL wrote the manuscript with support from S-NL. QL and S-NL developed the theoretical formalism, performed the analytic calculations. QL performed the numerical simulations. X-LD assisted with sample and supervised the project. All authors conceived of the presented idea, discussed the results and contributed to the final manuscript. All authors read and approved the final manuscript.

Authors’ Information

Quan Liu, born in 1992, is currently a postgraduate at Robotics Institute, School of Mechanical Engineering and Automation, Beihang University, China. His main research interests include robotics, pointing mechanism and machine design. E-mail: liuquan2015@buaa.edu.cn.

Sheng-Nan Lu, born in 1987, is currently a postdoctoral researcher at Robotics Institute, School of Mechanical Engineering and Automation, Beihang University, China. Her research interests include kinematic analysis, design of deployable mechanisms, analysis of reconfigurable robots. E-mail: lvshengnan5@gmail.com.

Xi-Lun Ding, born in 1967, is currently a full professor at Robotics Institute, School of Mechanical Engineering and Automation, Beihang University, China. His main research interests include space mechanisms, kinematics, dynamics, design and control of robotics. E-mail: xlding@buaa.edu.cn.

Competing Interests

The authors declare that they have no competing interests.

Ethics Approval and Consent to Participate

Not applicable.

Funding

Supported by National Natural Science Foundation of China (Grant Nos. 51635002 (Key Program), 51605011, 51275015).

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Authors’ Affiliations

(1)
School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191, China

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