Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 2 (September 2016), 79-83

Modeling and Control of a Suspended Gravity Compensation Systemwith Rigid-Elastic Coupling
Jiao Jia

The Seventh Research Division and the Center for Information and Control, School of Automation Science andElectrical Engineering, Beihang University (BUAA)
Beijing, 100191, China
Yingmin Jia

The Seventh Research Division and the Center for Information and Control, School of Automation Science andElectrical Engineering, Beihang University (BUAA)
Beijing, 100191, China
Shihao Sun

The Seventh Research Division and the Center for Information and Control, School of Automation Science andElectrical Engineering, Beihang University (BUAA)
Beijing, 100191, China
E-mail: annebuaa@hotmail.com; ymjia@buaa.edu.cn; jxcrssh@126.com
www.buaa.edu.cn

Abstract
In this paper, a suspension gravity compensation system (SGCS) is modeled and the corresponding controller isdesigned. The system is a servo platform consisting of three sub-systems by which a micro-gravity environment canbe established. Especially, a novel zero stiffness suspension sub-system and a creative unconstrained structure areproposed. The system model is deduced based on Lagrange equation. A PID controller is designed by using thefeedback linearization approach. Simulation results show the effectiveness of the proposed method.
Keywords: gravity compensation, rigid-elastic coupling, Lagrange method, feedback linearization, PID

1. Introduction
One of the vital differences between the spaceenvironment and the ground laboratory environment isthat the space is micro-gravity. In order to represent thespacecraft motion in space on the ground and improve
the fidelity of ground verification of guidance andcontrol system, the micro-gravity environment should
be built. Up to now, the gravity compensation methods
have been developed such as weight loss, neutral
buoyancy, air-bearing and suspension system1-5 .There

are both advantages and disadvantages of theseapproaches. Among them the suspension method5-7
becomes an economical, practical and reliability
method.
There are two kinds of suspension compensation
methods passive suspension compensation5 and active
suspension compensation6-7. The active suspension
method performs much better than the passive one. The
compensation accuracy of the suspend system depends
on the structure and controller of the vertical direction
sub-system greatly. Usually, a wire rope is one of thePublished by Atlantis Press
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79

Jiao Jia et al.

necessary parts of the vertical direction sub-system.
However, when a force which is opposite to or greaterthan the object gravity functions on the object, the ropeis invalid. To solve this problem, a spring is combinedwith the rope, but the compensation accuracy is stillaffected by the rope reversing. In this paper, we develop
a new means that combines a spring buffer unit with thetransmission unit to avoid the shortage of the wire rope.
Besides, there are rarely papers or researches thatconcern how to guarantee the object rotation freedoms
when it is suspended to the compensation platform. In
this paper, a creative mechanical structure is presented.

mechanism and they are driven by the motor_2 and the
motor_1 respectively. The X direction mechanism fixes
on the supporting frame.
The USS consists of twelve bearings and their
fixations as presented in Fig.2. The bearings areclassified into yaw bearings, roll bearings and pitch
bearings according to their corresponding support
rotation of the object.

2. System Design
The platform is a servo system consisting of a levelservo sub-system (LSS), a zero stiffness suspension subsystem (ZSSS), an unconstrained suspension sub-system(USS) and a circuit control section (CCS). The LSS canfollow the object’s horizontal movement and maintainthe ZSSS vertically. The ZSSS could compensate thegravity of the object suspended to the USS and assure
its gravity wouldn’t affect its movement. The USS holds
the object and such that the object would rotate freely.
Note that they work cooperatively to create a microgravy environment.
As described in Fig.1.the object connects to theplatform through the USS. The USS is composed oftwelve rolling bearings and their fixations, whichguarantees the equivalent suspension point coincides
with the object centroid. Hence, the object can keepbalance at any random attitude. The USS links to thespring buffer. The spring buffer attaches to the ZSSS.
The ZSSS includes a universal joint, the Z direction
mechanism and the motor_3. The CCS contains a dualaxis tilt sensor, a tension sensor, drive control cards, anacquisition card and a PC. The dual-axis tilt sensor isfixed to the spring buffer through an aluminum panel.
The spring buffer and the tension sensor are connectedby the universal joint. Therefore, when the object
moves, the spring buffer swings around the universal
joint. The dual-axis tilt sensor can measure the swing
angle. The tension sensor connects to the Z direction
mechanism. The Z direction mechanism transmits by arack and a pinion driven by the motor_3. The ZSSS
installs on the LSS specifically on the Y direction
mechanism. In addition, the LSS comprises themotor_2, the X direction mechanism and the motor_1.
The Y direction mechanism mounts on the X direction

Fig.1.Structure and coordinate system of SGCS

The working principle of the platform is as follows.
When the object moves along the horizontal plane thedual-axis tilt sensor can measure the swing angles. Thenthe motor_1 and motor_2 can drive the X and Y
direction mechanism to eliminate the swing angle tomaintain the spring buffer vertically. When the object
moves vertically the length of the spring will vary. Thetension sensor will detect the variation and then themotor_3 actuates the Z direction mechanism toeliminate the variation. So the vertical movement of theobject wouldn’t be affected by its own gravity. Theobject is in a micro-gravity environment.

Fig.2. Structure of USS

3. Model Building
The SGCS coordinate system is shown in Fig.1.right.
Among it, the spring buffer and the application of USS
are equivalent to a spring and a rope linked to thecentroid of the object respectively. The meaning of eachpart is as below.

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Suspended Gravity Compensation Systemm0 , the load of LSS,

m1 , m2 , the mass of the pinion and rack resp.,
M , the mass of object,
α , the rotation angle of the pinion
Q − XYZ , static coordinate systemO , centroid of m0O1 , O2 , O4 , centroids of m1 , m2 , M resp.,

O3 , centroid of the universal joint,






Fx , Fy , motor_1&2 equivalent driving force on

X&YTx , motor_3 output torque on the pinion
F4 x , F4 y , F4 z , the object driving force

T4 x , T4 y , T4 z , the object driving torqueincluded in 0.5m0 x vm2 0 x + 0.5mm 0 y v y2 ( m0 x , m0 y the load ofX, Y direction resp.).
The system kinetic energy isT=Choose xOy plane as zero potential energy surface
and the system potential energy is=
V m2 gα R + 0.5k (l + l1 ) 2 + Mg (α R − 0.5h0 − d )(1)d  ∂L  ∂L  − = Q j ( j= 1, 2,......)
dt  ∂q  ∂q(2)Substitute Eq. (1) into Eq. (2) and define,M + m0 y
M + m0 x
1,η y =
m1 + m2 + M ,η x =2MM
m2 + M
1mk,η 2,ς,ρ=η===MMMM222yβ =+β β ,β =β β +β βm=

O3 − x31 y31 z31 , coordinate rotate along the buffer.
From the definition above, the coordinates ofO, O1 , O2 are ( x, y, 0 ) , ( x, y − R, 0 ) and ( x, y, α R ) .

And the velocity of each point is ( x , y , 0 ) , ( x , y , 0 )and ( x , y , α R ) .

L= T − V

Lagrange equation

β ,wing angle between spring buffer & vertical
β x , β y , orthogonal decomposition of βO − xyz , coordinate moves with O ,Oi − xi yi zi , i =3, 4 , coordinates move with O3 & O4x

Define l1 = Mg

, d = l0 + l1 + l . Then O4 isk( x + d sin β x cos β y , y + d sin β y , α R − 0.5h0 − d cos β x cos β y )h0 is the length of the rack. O & O2 coincide at theinitial position. l is the variation length of the spring. Setl = 0 , when the force of the spring is equal to Mg .
For β ≤ 5 , sin β ≈ β , cos β ≈ 1 , then

( x + d β , y + d β ,α R − 0.5h − d )xy0

The velocity of O4 isx + lβ x + d βx , y + lβ y + d β y , α R − l

(1 111( m1 R 2 )α 2 + m2 (α R) 2 + m0 x vm2 0 x2 22211222+ mm 0 y v y2 + M (vMx+ vMy+ vMz)
22)

The attitudes of the object will be changed as longas one of the T4 x , T4 y , T4 z is valid, and therefore theangular velocity is time-varying. But when we choosex, y, α , l , β x & β y as the system generalized coordinates,
they are independent of the object rotational kineticenergy. Hence, we could ignore it when we calculate thesystem kinetic energy.
We define m0 as the load of LSS. The pinion andrack move along the X&Y direction mechanism at thehorizontal direction, so this part of kinetic energy isyxxxyy

The equations deduced from Eq. (2) can beformatted in a compact form
η x
0

0
βx
d
 0
0

0
0

00

00βx
βy

d00

ηy000

η R2−Rβy0−R0d0

β 2 +1 d βxd βxd20dβy0 00 02 βx
2 β y0 00 002 β xy0 0 2d βx0 0 2d β y0   xd   y 0   α  +d β y   l 0β   x 2d   β y 0 0   x   0 
 Fx    F 
0 0  y   0 
 y
 Tx αη g R 0 0
  + 2ρ 
=0 0  l   z l  F4 z  F4 x 
0 0 βx0   

0 0   β y   0  F4 y (3)We classify the system generalized coordinates into
two parts based on their controllability that x, y, α only
controlled by the motors and the other ones are affectedby the driving forces of the object simultaneously.

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Jiao Jia et al.
TβxTβ y  , andT(5)M 2 q1 + M 3 q2 + C3 q1 + C4 q2 + G2 =F2Obviously, M 2 is invertible
Let Eq. ( 4 ) − M 1 M 2−T Eq. ( 5 ) . And C=
C=0 we1
3gain(6)M m q2 + Cq2 + G = F1 − M 1 M 2 F2Mm =M 2 − M 1 M 2 −T M 3 , C =C2 − M 1 M 2 − T C4 ,Output of l
Output of bitax
Output of bitay

20.2
0-0.2-0.4

10-1-0.6-2-0.8
15time/s

1050-330

25

200System Output

0.3F4 zmovement requirements and is uncontrollable for SGCS,
it’s regarded as interferences. So the model is anonlinear time-varying
coupling modelwithdisturbances8-10. A PID controller is designed by using
the feedback linearization approach.

F=
F11 + M m F121(7)=F11 Cq2 + G(8)(F12 =− K P q2 + K I ∫ q2 + K D q2)

=K P =K Pz K Px K Py  , K I  K Iz K Ix
K D =  K Dz K Dx K Dy (9)K Iy 

F11 is used to counterweigh Cq2 + G of Eq. (6).
Substitute Eq. (7) in Eq. (6), and we gain−T M m F12 − M 1 M 2 F2
M=m q2(10)

It’s easy to verify the existence of the inverse ofM m . Eq. (10) simplifies toq2 =F12 + d (t), d (t) =− M 1 M 2 −T F2(11)Where F12 is a PID controller. PID parameters
adjustment rules are employed to find suitable control
parameters for the system.
5. Simulation Results
Firstly, the control target is q2 = 0 . It means the gravityof the object is compensated completely.

20-32530

System Output Error

x 10

Output Error of l
Output Error of bitax
Output Error of bitay

10
Output Error Signal

Output Signal

F4 z ] depends on the object

15time/s

2Output of l
Output of bitax
Output of bitay

0.2

Since F2 = [ F4 z

10The object driving input is shown in Fig.3.left whichis regarded as the system disturbance. From Fig.3.right,
the system spring variation error is less than 0.0015 m
which means the compensation accuracy is up to 99%.
0.25

T5Fig.3. The object driving input and output of the system= G1 − M 1 M 2 −T G2
G
4. Controller Design

System Output

x 10

3

0.4

-1−T4
Z Direction
X Direction
Y Direction

0.6(4)

M 1q1 + M 2 q2 + C1q1 + C2 q2 + G1 =F1-3

Object Driving Input

10.8Output Signal

x y α ] , q2 l[=Object Driving Signal=Define q1

0.15
0.1
0.05

-1-2-3-4

0-5-0.05

05

1015time/s

202530

16.2

16.3

16.4

16.5

16.6
time/s

16.7

16.8

16.9

17Fig.4. The system response & output error of the 0.2step input

Then set q2 = [u (t) 0 0] . u (t) is the reference
input of the system which means when the spacecraft is
not complete weightlessness and it varies according to
the law role of the gravity effect.
Fig.4. shows the system response and output error at0.2step reference input with the same disturbance asshown in Fig.3.left. The compensation accuracy is up to99%. The simulation results show that smaller of spring
stiffness coefficient the higher compensation accuracy.
T

6. Conclusions
In this paper, a novel compensation platform reflecting
in the ZSSS and USS is proposed and then themathematic model based on Lagrange equation isdeveloped. Lastly, a stable and high compensation
accuracy controller is given. However, the existence ofthe sensor errors and modeling errors, we can’t
compensate the nonlinear parts completely. The depth
study should be conducted to find more practicalcontrollers11-15.
Acknowledgements
This work was supported by the National Basic
Research Program of China (973 Program:
2012CB821200, 2012CB821201) and the NSFC
(61134005, 61327807, 61521091, 61520106010).

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