 Original Article
 Open Access
An Error Equivalent Model of Revolute Joints with Clearances for Antenna Pointing Mechanisms
 Quan Liu^{1},
 ShengNan Lu^{1}Email authorView ORCID ID profile and
 XiLun Ding^{1}
https://doi.org/10.1186/s1003301802336
© The Author(s) 2018
 Received: 29 June 2017
 Accepted: 16 April 2018
 Published: 6 May 2018
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 dualaxis (“X–Y” type) antenna pointing mechanism of Spotbeam 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 realtime tracking and precise pointing on target satellites, dualaxis antenna pointing mechanisms have been widely applied in communication satellites and data relay satellites for satelliteground and satellitesatellite 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 dualaxis 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 [3–6]. Three main methods dealing with clearance in revolute joints are proposed, the springdamper 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 “contactseparation” twostate model, Li et al. [12] established the multibody system dynamic equations of twodimensional pointing mechanism with clearance. Zhang et al. [13] studied the comparison on kinematics and dynamics between the fully actuated 3RRR mechanism and the redundantly actuated 4RRR 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 ParentiCastelli [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 computeraided 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 [23–29], 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 outofplane 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. [20–22] and it is easier to be applied to analysis of pointing accuracy of the spatial dualaxis pointing mechanism (Additional file 1).
2 Modeling of Revolute Joints with Clearances
2.1 Revolute Joints with Clearances
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 O_{1} 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, k_{max}]. 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 Oxy, 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 crosssections of the journal on two end faces of the bearing can be different which generates different positions and orientations 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 Oxz plane or Oyz plane. General distribution of the journal is illustrated in Figure 2(d).
2.2 EEM of the Revolute Joint with Clearances
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 Ox_{1}z_{1} 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
Therefore, \(O_{3}^{1}\) = [k_{2}cosθ_{2}, k_{2}sinθ_{2}, − l]^{T}.
As shown in Figure 3, x_{5} axis is perpendicular to Plane 2 and z_{5} axis is along the direction of \(l_{{O_{3} O_{4} }}\).
 Step 1::

Translate the O_{1}x_{1}y_{1}z_{1} coordinate system to O_{5};
 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
3.2 Constraint Analysis
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 Dualaxis (“X–Y” 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.
In Figure 5(a), O_{11} is the midpoint of segment \(l_{{O_{9} O_{10} }}\); O_{7} and O_{8} express the center of revolute joints in the revolute joint Y, respectively. O_{6} is the midpoint of segment \(l_{{O_{7} O_{8} }}\). Meanwhile, O_{9} and O_{10} 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 O_{7}. 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.
 Step 1::

Translate O_{5}x_{5}y_{5}z_{5} with a distance, k_{5}, along x_{5} axis.
 Step 2::

Translate the coordinate system obtained in Step 1 with a distance, l/2, along y_{5} axis.
 Step 3::

Rotate the coordinate system obtained in Step 2 with an angle, − π/2, around its x axis.
5 Case Studies
Parameters of the EEM in case studies
Parameter  Case 1  Case 2  Case 3 

\(\Delta l\) (mm)  0  0.01  0.1 
k_{1} (mm)  [0, 0.05]  [0, 1]  [0, 1] 
k_{2} (mm)  [0, 0.05]  [0, 1]  [0, 1] 
θ_{1} (rad)  [0, 2π]  [0, 2π]  [0, 2π] 
θ_{2} (rad)  [0, 2π]  [0, 2π]  [0, 2π] 
l (mm)  180  180  180 
5.1 Case 1
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 d_{max} = k_{1max} = 0.05 mm.
5.2 Case 2
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 2RPUC 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 subcases.
It can be seen that the maximum value of k_{2} reduces to 0.897 mm which indicates the k_{2} can only range from 0 to 0.897 mm.
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.
5.3 Case 3
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 2RPUC mechanism, is established. Then, the model is extended for describing the error of a spatial dualaxis (“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 SNL. QL and SNL developed the theoretical formalism, performed the analytic calculations. QL performed the numerical simulations. XLD 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. Email: liuquan2015@buaa.edu.cn.
ShengNan 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. Email: lvshengnan5@gmail.com.
XiLun 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. Email: 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
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