An Error Equivalent Model of Revolute Joints with Clearances for Antenna Pointing Mechanisms
Liu et al. Chin. J. Mech. Eng.
An Error Equivalent Model of Revolute Joints with Clearances for Antenna Pointing Mechanisms
Quan Liu 0 1
Sheng‑Nan Lu 0 1
Xi‑Lun Ding 0 1
0 School of Mechanical Engineering and Automation, Beihang University , Beijing 100191 , China
1 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:
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.
Error modeling; Joint clearances; Antenna pointing mechanism; Radial clearance; Axial clearance
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 [
]; 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 [
]. Three main
methods dealing with clearance in revolute joints are
proposed, the spring-damper approach, the massless link
approach and the contact force approach . Wang et al.
] 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. [
] 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.
] proposed a simple approach to identify the largest
position and direction errors. Zhang et al. [
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. [
established the multibody system dynamic equations of
two-dimensional pointing mechanism with clearance.
Zhang et al. [
] 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. [
] presented the uncertainty analysis
of robots with revolute joint clearances, which can be
applied in both planar and spatial mechanical systems.
Bai et al. [
] established a hybrid contact force model
to forecast the dynamic performance of planar
mechanical systems with revolute joint clearances. Venanzi and
] 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.
] presented a method to model planar and spatial
joints with clearances. Brutti et al.  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. [
] 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. [
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
], 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. [
] 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 [
]. In the
cross-section, the difference between radii of the bearing
and the journal defines the range of radial clearance.
In Figure 1(a), the radial clearance can be described as
0 ≤ k ≤ kmax,
kmax = Rb − Rj,
where Rb and Rj are radii of the bearing and the journal,
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, lOO1, 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
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(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.
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 lO3O4. 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. lO′ O′ and lO4′O4 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 lO4′O4 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,
Figure 3 An EEM of the revolute joint with clearances
x1 axis is perpendicular to the reference plane. The O
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 lO3O4. In
Figure 3, ki (i=1, 2) is defined as the translational distance
of the ith prismatic joint, ki∈[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.
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
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
Oij = [xOi , yOi , zOi ]T is the coordinate of Oi in Oj-xjyjzj.
Direction of the ideal journal is assumed to be [
0, 0, 1
O1-x1y1z1. It is obvious that,
O2= [k2, 0, 0]T,
O1= [k1 cos θ1, k1 sin θ1, 0]T.
Coordinate of O3 in O1-x1y1z1 can be calculated as
Therefore,O31 = [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
k1 cos θ1 − k2 cos θ2
n=O41 − O1=
3 k1 sin θ1 − k2 sin θ2 .
As shown in Figure 3, x5 axis is perpendicular to Plane
2 and z5 axis is along the direction of lO3O4.
p = (cos α, 0, sin α).
q = n × p.
O′31 = [k1 cos θ1, k2 sin θ2, −l]T,
O′41 = [k1 cos θ1, k2 sin θ2, 0]T,
(k1 cos θ1 + k2 cos θ2)/2
O51= (k1 sin θ1 −+l/k22 sin θ2)/2 .
tan β = lO4O4′ /lO3O4′ ≈ lO4O4′ /lO3′O4′
= (k2 sin θ2 − k1 sin θ1)/l,
α = arctan[(k2 cos θ2 − k1 cos θ1)/l],
β = arctan [(k2 sin θ2 − k1 sin θ1)/l].
cos α 0 − sin α 0
0 1 0 0
Ry(−α)= sin α 0 cos α 0 ,
0 0 0 1
1 0 0 0
0 cos β − sin β 0
Rx(β) = 0 sin β cos β 0 ,
0 0 0 1
Meanwhile, O5-x5y5z5 can be constructed by the
following three steps:
Translate the O1-x1y1z1 coordinate system to
Rotate the coordinate system produced in
Step 1 with an angle, − α, around its y axis;
Rotate the coordinate system produced in
Step 2 with an angle, β, around its x axis
Then we can get
where 15T defines the homogeneous transformation
matrix from O5-x5y5z5 to O1-x1y1z1,
tan α = lO3O3′ /lO3′O4′ = (k2 cos θ2 − k1 cos θ1)/l,
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
Then direction of y5 axis can be calculated as
Therefore, As shown in Figure 3, we can gain the following equations,
where a = (k1 cos θ1 + k2 cos θ2)/2,b = (k1 sin θ1 + k2 sin θ2)
/2, cα = cosα, sα = sin α, cβ = cosβ, sβ = sin β.
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 lO3O4 =l + l. l is the distance
between O1 and O2, as well as the distance between
O3 and O4 along with z1 axis. 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,
k12 + k22 − 2k1k2 cos(θ1 − θ2) + l2.
When θ2=θ1+π, the distance reaches its maximal
Herein, we define l′ as,
l′=lO′3O4max − l.
When l ≥ l′, the physical journal can freely carry
out the translational and rotational motion constrained
within the boundary of bearing. However, if l < l′,
movement of the physical journal will be limited.
4 EEM of the Dual‑axis (“X–Y” Type) Antenna
Pointing Mechanism with Revolute Joint
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
Plane 3 is parallel to Plane 2, it also passes O7. In
addition, lO5O6is perpendicular to both Plane 2 and Plane 3.
lO7O8 and lO3O4 are orthogonal to each other.
Similarly, we can get the homogeneous transformation
matrix from O11-x11y11z11 to O7-x7y7z7,
the revolute joint Y is orthogonal to the physical journal
of the revolute joint X.
In Figure 5(a), O11 is the midpoint of segment lO9O10; O7
and O8 express the center of revolute joints in the
revolute joint Y, respectively. O6 is the midpoint of segment
lO7O8. Meanwhile, O9 and O10 are defined as the center
of the U type hinges in the equivalent model of joint Y.
O75 = [k5, l/2, 0]T.
5.1 Case 1
In the first case, l = 0 mm, it means that the revolute
joint has only radial clearance and no axial clearance.
l = 0 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.
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
l = 0.01 mm in the second case. From Eqs. (15) and (16),
by using the parameters in Table 1, the following can be
lO′3O4max = 180.011 mm,
l′=lO′3O4max − l = 0.011 mm.
It is obvious that l < l′, which limits the movement
of the physical journal. The radial clearance is restricted
by the axial clearance which results in the restrains of ki
and θi. 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.
Herein, ly−O3O4 is the distance between O3 and O4
projected on O1-x1y1 plane:
l)2 − l2,
k2′max = ly−O3O4 − k1max.
Therefore, ly−O3O4 = 1.897 mm,
k2′max = 1.897 − 1=0.897 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
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.
ϕ = arccos (k12max + k22max − ly2−O3O4 )/2k1maxk2max .
It can be calculated that ϕ=0.795rad, 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
ψ max = arctanly−O3O4 /l = 0.604◦,
dmax = k1max = 1 mm.
For easier understanding, a possible configuration of
the journal in case 2 is described in Figure 9.
5.3 Case 3
In the third case, we have l ≥ l′, 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.
The maximum distance and angle in case 3 is
ψ max = arctan[ (k1max + k2max)/l]=0.637◦,
Through the above analysis, it can be known that when
l = l′, 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
l = l′.
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.
Additional file 1. Brief introduction of the paper.
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
The authors declare that they have no competing interests.
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