Journal of Robotics, Networking and Artificial Life
Vol. 8(2); September (2021), pp. 94–98
DOI: https://doi.org/10.2991/jrnal.k.210713.005; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Adaptive Sliding Mode Control of Inclination Angle
and Tension for a Suspended Anti-Gravity Device
Yuxin Jia, Yingmin Jia*, Kai Gong
The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering,
Beihang University (BUAA), Beijing 100191, China

ARTICLE INFOABSTRACT

Article History

In this paper, the Suspended Anti-Gravity Device (SAGD) is studied to construct sustainable microgravity environment on the
ground. To overcome the difficulty of accurate gravity compensation when SAGD is disturbed, a new adaptive sliding mode
controller is proposed based on the dynamic model, which effectively ensures the stability of inclination angle and tension. The
simulation and experiment results show that the transient time and steady-state error are small, which satisfies the requirementsof spacecraft ground verification.

Received 06 November 2020
Accepted 08 May 2021
Keywords
Suspended anti-gravity device
microgravity
adaptive sliding mode control
ground verification© 2021 The Authors. Published by Atlantis Press International B.V.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
In 2020, space probes such as “Perseverance”, “Chang’e-5” and“Tianwen-1” have been launched to explore Mars and the moon.
It shows that human beings are marching toward deep space, ushering in a new era of rapid development of space technology. As
we all know, it is difficult for human beings to reach space, which
determines that the development of space technology is particularly dependent on ground experiments [1–3]. Since microgravity is the most important feature of space environment, many
important devices have been developed to construct microgravity
environment on the ground, including drop tower, air-floating
platform, Suspended Anti-Gravity Device (SAGD), etc. However,
the operation time and internal movable space of the drop tower
and air-floating platform are limited. The SAGD offset the gravity of test object through the tension of hanging wire, breaking
through the above limitations [4,5]. In the 1990s, Carnegie Mellon
University developed an experimental device to verify various
algorithms of space manipulator. The device uses counter-weight,
pulleys and other mechanical structures to generate constant tension on the steel wire to offset the gravity of space manipulator.
Subsequently, SAGD is concerned by many scientific research
institutions [6]. Harbin Institute of Technology used single slingto offset five-sixths of the gravity of “YuTu” lunar rover, and measured its wheel pressure in a simulated lunar gravity environment.
In 2019, Beijing Institute of Space Electromechanical Researchdeveloped a landing test device to verify the performance of probe
during the landing phase in extraterrestrial body. It has been successfully applied to the landing test of Mars probe. The height of theCorresponding author. Email: ymjia@buaa.edu.cn*

device is 140 m, and the diameter of the experimental area is 120 m.
By adjusting the tension of 36 steel wires, it can simulate the gravity
environment of extraterrestrial body.
The SAGD has been widely applied. Therefore, it is significant tostudy how to keep the inclination angle and output force constant in
the presence of disturbance. An Adaptive Sliding Mode Controller(ASMC) is proposed in this paper. Compared with the traditional
method, the anti-interference ability of SAGD is improved.

2. SYSTEM MODELINGThe SAGD consists of a horizontal servo system, a low stiffness
structure, an unconstrained structure, a universal joint, a suspension system, and a test satellite, as shown in Figure 1 [5].
When the test satellite is disturbed, the horizontal servo system
can quickly track its horizontal movement, and ensure that the low
stiffness structure remains vertical. The universal joint and unconstrained structure can ensure that the freedom of test satellite is
not restricted, so that the test satellite can perform approximate
unconstrained pitching, yawing and rolling motion under a small
frictional torque disturbance. The suspension system can control
the output tension of SAGD by adjusting the moving distance ofrack. A tilt sensor is installed under the universal joint to measure
the inclination of low stiffness structure. Only when the inclination
angle is 0 and the output force is equal to the gravity of test satellite, can accurate gravity compensation be realized and a long-term
microgravity environment can be constructed. How to establish
the coordinate system, please refer to Figure 2. O1 is the center ofthe gear in suspension system, O2 is the meshing point with rack.

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The matrices defined in the model are as follows:ùé -w x (h1 - 1)
-d(h1 - 1)
0
H = ê -w y (h2 - 1)
-d(h2 - 1)ú
0
ú
ê
0
0
ûú
ëê - R(M - Mc ) M
ù
é 0
0
-h1 d
ú
 =ê 0
H
0
-h2 d
ú
ê
ëê RM c M -w x RMc (Md ) -w y RMc ( Md )úû
éF ù
éd ù
é -2w x (h1 - 1) 0 0ù
1 x
1 z
J = ê -2w y (h2 - 1) 0 0ú , F = ê Fy ú ,c = êd x úúêMê úM êd ú
00 0ûúëêTa ûú
ëê y ûú
ëê
é
ù
éLù
0
ú , q = êw ú
N=ê
0
ê
ú
ê xú
2
ëêw y ûúëê R( g + (MMb g + LMc k) M )ûú

Figure 1 | Schematic diagram of SAGD.

d = L + L0 + Mg k , Mc = Ma 2 + Mb + M
h1 = (M + M1 ) M , h2 = (M + M2 ) M
where F is the input of SAGD. q is the state vector, including
the variation length of low stiffness structure and the orthogonal decomposition values of w. d is the disturbing forces and isassumed to be bounded.

3. ASMC DESIGN
Figure 2 | The coordinate system of SAGD.

O3 and O4 are the center of universal joint and test satellite respectively. The physical meanings of the main symbols are as follows:•• g, Gravitational acceleration.
•• R, Radius of the gear in suspension system.

In this section, a new ASMC is designed, which can reduce theinfluence of uncertain disturbing forces on the accuracy of gravity
compensation. Define variables:A(q, q ) = − H −1 ( Jq + N )T c = c , c , c 
B(q) = H −1 , c = H −1H
 1 2 3The dynamic model (1) is transformed into•• k, Elastic coefficient of low stiffness structure.

		 q = A(q, q ) + B(q)F + c •• w, Inclination angle of low stiffness structure.

(2)•• M, The sum of the masses of unconstrained structure and test
satellite.

To offset the gravity of test satellite, and verify its attitude control
performance in floating state, this paper defines the expected valueof state vector q as qd = [0,0,0]T. The system error is e = q – qd , andthe upper bound of c is a = [b1, b2, b3]T. Thus, Equation (2) istransformed into•• Ta, Driving torque of the motor in suspension system.

	    e = A (q, q ) + B (q) F + c − qd 

•• M1, M2, The loads on X- and Y-direction motor of horizontal
servo system.

The switching function is defined as:•• L0, Initial length of low stiffness structure.
•• L, Variation length of low stiffness structure.

•• Ma, Mb, The masses of the gear and rack in suspension system.
•• Fx, Fy, Driving forces of X- and Y-direction motor of horizontal
servo system.
•• dx, dy, dz, Disturbing forces.
•• wx, wy, Orthogonal decomposition values of w .
According to the operation principle of SAGD, the model can beobtained by Lagrange equation.
d 		 Hq + Jq + N = F + H(1)

s = De + Vsgn(e )z

T
  
z
z
z
sgn(e )z = é e1 sign(e1 ), e2 sign(e2 ), e3 sign(e3 )ù
ë
û

(3)

(4)

where, D = diag (d1, d2, d3) and V = diag (v1, v2, v3) are positive
definite matrices, s = [s1, s2, s3]T, z = diag (z, z, z) and satisfies
1 < z < 2.
The adaptive parameter φ = [φ1, φ2, φ3]T is used in controller design
to improve the anti-interference ability of SAGD. The adaptive law
is defined as Equation (5), and φ is dynamically adjusted according
to the value of switching function Equation (4).

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Té s1 s2 s3 ù, ú 		 i = ê ,ë f1 f 2 f 3 û(5)where, f1, f2, f3 are all positive constants.
T
Assumption 1. The stable value i = éj 1 , j 2 , j 3 ù of the adaptiveëûparameter satisfies j p > b p , p = 1, 2, 3.
Theorem 1. Under Assumption 1, the system error e will converge
to 0, if the ASMC F is designed as:
F = Fe + Fd + Fs

1
1
1
æ
ö
,
,

Fe = -B(q)-1 diag ç
z -1
z -1
z -1 ÷ De



zv
e
zv
e
zv
e

è 1 1
ø
2 2
3 3
-1
Fd = B(q) (- A(q, q ) + qd )Fs = - B(q)-1 (diag(i ) sign (s))
When ėp = 0, define(

1 zv p e p

z -1

(6)

to the system. i is used to represent the initial value of φ, andthe parameters of ASMC are as follows:
f1 = f 2 = f 3 = 1, i = [1, 1, 1]T , z = 1.1
D = diag(280, 510, 450), V = diag(7.5, 8, 9)
The simulation results are shown in Figures 4 and 5. It can be foundthat the maximum errors of L, wx and wy are 5.151 × 10−5 m, 3.779 ×
10−5 rad, 1.254 × 10−5 rad, respectively. Therefore, the output tension error is 0.036 N, and the gravity compensation accuracy isabout 99%. In the following, the ASMC is used to compare withthe controller in Jia et al. [5]. By analyzing the results in Figures 4
and 6, it can be concluded that the transient time of the ASMC
system is shorter and the steady-state error is smaller. When thesimulation time t >1.6 s, the state error of ASMC system satisfies||e|| < 4 × 10−6, while the control system in Jia et al. [5] needs 1.7 s.
Furthermore, on the basis of disturbance, Gaussian noise obeying) = 0, p = 1, 2, 3
Proof. The error of adaptive parameter i is i = i - i , and the
Lyapunov function is defined as:
1 T
1s s + i diag( f1 , f 2 , f 3 ) diag (q 1 , q 2 , q 3 ) i
2
2
  

z -1
diag(q 1 , q 2 , q 3 ) = zVdiag e
VL =

( )

(7)

Furthermore, the derivatives of Equations (4) and (7) can beobtained( )Figure 3 | Disturbing forces.

z -1s = De + zVdiag e
e

T
T
VL = s s + i diag ( f1 , f 2 , f 3 ) diag(q 1 , q 2 , q 3 ) i

(8)

Substitute the ASMC Equation (6) into Equation (8)

(

) (
)

)

VL = q 1 c1 s 1 - j 1 s 1 + q 2 c2s 2 - j 2 s 2+ q c s - j s £ 0
3

(

3 3

3

3

(9)

According to the analysis of Equation (9), the ASMC can stabilize the system. Substituting the controller Equation (6) into thesystem error model Equation (3), it can be found that the solution(e, ė) of the system error model will not make VL ≡ 0 except fororigin (0, 0). Therefore, the system error e will converge to 0. The
proof is completed.

Figure 4 | State response of the ASMC system.

4. SIMULATION ANALYSIS
In this section, to verify the ability of ASMC, the simulation experiment is carried out. Equation (10) is used instead of the sign function Sign(s) to reduce the oscillation of state response.

()

s s + 0.05 
		  

(10)

The parameters of the SAGD are R = 0.02 m, M = 15.5 kg,
Ma = 0.52 kg, Mb = 4 kg, M1 = 16 kg, M2 = 41.5 kg, L0 = 0.45 m,
k = 690 N/m. The pulse disturbance shown in Figure 3 is applied

Figure 5 | The output of ASMC.

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N(0,1) is added. The simulation results are shown in Figure 7, thestate error satisfies ||e|| < 6.5 × 10−4, which still satisfies the requirements of ground verification.

5. EXPERIMENT RESULTS
To further verify the superiority of ASMC proposed in thispaper, experiments were carried out in the actual device shownin Figure 8. The experiment results are shown in Figures 9 and10. The maximum errors of wx and wy are 0.55° and 0.18°, whichmeans that the low stiffness structure remains approximately vertical. The expected value of force sensor output is Mg = 151.9N,

Figure 9 | The output of tilt sensor.

Figure 10 | The output of force sensor.
Figure 6 | State response of the control system in Jia et al. [5].

and the maximum error is 9.1N. Therefore, SAGD can compensate the gravity with an accuracy of about 94%.

6. CONCLUSION
In this paper, the dynamic model of SAGD is established by
Lagrange equation, and then a new ASMC is proposed. The controller can make the system asymptotically stable, and overcome
the difficulty of accurate gravity compensation in the presence ofdisturbance. The simulation results show that, compared with theexisting methods, the ASMC has the advantages of small steadystate error and fast convergence speed. In addition, the performance of ASMC has been tested through the actual device, which
can satisfy the requirements of ground verification.
Figure 7 | State response of the ASMC system with noise.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.

ACKNOWLEDGMENT
This work was supported by the NSFC (61327807, 61520106010).

REFERENCES[1][2]
Figure 8 | Suspended anti-gravity device.

Z. Jiang, J. Xu, H. Li, Q. Huang, Stable parking control of a robot
astronaut in a space station based on human dynamics, IEEE
Trans Robot 36 (2020), 399–413.
C. Wang, H. Nie, J. Chen, H.P. Lee, The design and dynamic analysis of a lunar lander with semi-active control, Acta Astronaut.
157 (2019), 145–156.

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[3]
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X.F. Liu, G.P. Cai, M.M. Wang, W.J. Chen, Contact control for
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A. Witze, NASA rethinks approach to Mars exploration, Nature
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[5]
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J. Jia, Y. Jia, S. Sun, Continuous non-singular fast terminal sliding
mode control for an active gravity field simulator, J. Robot. Netw.
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AUTHORS INTRODUCTION
Mr. Yuxin Jia

Mr. Kai Gong

He received the B.S. degree in automation from Chongqing University, China,
in 2017. He is currently pursuing the PhD
degree with the Seventh Research Divisionand the Center for Information andControl, School of Automation Scienceand Electrical Engineering, Beihang
University. His current research interests
include gravity compensation and intelligent control of robots.

He received the B.S. degree in measurement & control technology and instrument from Harbin Engineering University,
China, in 2016. He is currently pursuing
the PhD degree with the Seventh Research
Division and the Center for Information
and Control, School of Automation Scienceand Electrical Engineering, Beihang
University. His current research interests
include motion and path planning for robotics and automation.

Prof. Yingmin Jia
He received the B.S. degree in control
theory from Shandong University, China,
in 1982, and the M.S. and PhD degrees
both in control theory and applications
from Beihang University, China, in 1990
and 1993, respectively. Then, he joined
the Seventh Research Division at Beihang
University where he is currently Professorof automatic control. He was the recipientof the National Science Fund for Distinguished Young Scholars
in 1996, and was appointed as Chang Jiang Scholar of the
Ministry of Education of China in 2004. His current researchinterests include robust control, adaptive control and intelligent
control, and their applications in robots systems.