Intelligent Control and Automation, 2011, 2, 112-120
doi:10.4236/ica.2011.22013 Published Online May 2011 (http://www.SciRP.org/journal/ica)
Reduced Model Based Control of Two Link Flexible
Space Robot
Amit Kumar1, Pushparaj Mani Pathak2, Nagarajan Sukavanam1
1Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India
2Department of Mechanical and Industrial Engineering, Indian Institute of
Technology Roorkee, Roorkee, India
Received January 16, 2011; revised April 11, 2011; accepted April 15, 2011
Abstract
Model based control schemes use the inverse dynamics of the robot arm to produce the main torque compo-
nent necessary for trajectory tracking. For model-based controller one is required to know the model pa-
rameters accurately. This is a very difficult task especially if the manipulator is flexible. So a reduced model
based controller has been developed, which requires only the information of space robot base velocity and
link parameters. The flexible link is modeled as Euler Bernoulli beam. To simplify the analysis we have con-
sidered Jacobian of rigid manipulator. Bond graph modeling is used to model the dynamics of the system and
to devise the control strategy. The scheme has been verified using simulation for two links flexible space
manipulator.
Keywords: Flexible Space Robots, Bond Graph Modeling, Reduced Model Based Controller, Euler-Bernoulli
Beam
1. Introduction
Space robots are a blend of a vehicle and a robot. It is
mechanically more complex than a satellite. The flexible
space robot will be useful for space application due to
their light weight, less power requirement, ease of ma-
neuverability and ease of transportability. Because of the
light weight, they can be operated at high speed. For
flexible manipulators flexibility of manipulator have
considerable influence on its dynamic behaviors. It is
observed by many researchers that fixed-gain linear con-
trollers alone do not provide adequate dynamic perform-
ance at high speeds for multi-degrees of-freedom robot
manipulators. Out of various schemes studied so far,
those involving the calculation of the actuator torque
(force) using an inverse dynamics model (the computed
torque methods), and those applying adaptive control
techniques have been extensively studied, and show the
greatest promise.
The traditional control strategy of robot manipulators
is completely error driven and shows poor performance
at high-speeds, when the high dynamic forces act as dis-
turbances. The current trend is towards model-based
control, where the dynamic forces are incorporated in the
control strategy as feed forward gains and feed-back
compensations along with the servo-controller which is
required only to take care of external noise and other
factors not included in the dynamic model of the robot.
As is expected, the model-based control scheme exhibits
better performance, but demands higher computational
load in real time. A particular area of model-based con-
trol is adaptive control, which is useful when the dy-
namic parameters of the robot are not well-known a pri-
ori. The controller adapts itself during execution of tasks
and improves the values of the dynamic parameters.
Murotsu et al. [1,2] proposed control schemes for
flexible space manipulator using a virtual rigid manipu-
lator concept. Samanta and Devasia [3] have discussed
modeling and control of flexible manipulates using dis-
tributed actuator. A technique is presented to analyze the
dynamics of flexible manipulators using bond graphs.
The nonlinear coupling of large rigid body motion and
small elastic vibration of the flexible arms has been taken
into consideration in the model. The concept of using
distributed piezoelectric transducers for controlling elas-
tic vibration of arms has been incorporated in the analy-
sis. Fardanesh and Rastegar [4] developed an inverse
dynamics model based on trajectory pattern method.

A. KUMAR ET AL.113
Wang and Gao [5] have reported inverse dynamics
model-based control for flexible-link robots, based on
modal analysis, i.e., on the assumption that the deforma-
tion of the flexible-link can be written as a finite series
expansion containing the elementary vibration modes.
Leahy et al. [6] worked on robust model based control.
They also discussed an experimental case. Fawaz et al.
[7] developed a model based real-time virtual simulator
of industrial robot in order to detect eventual external
collision. The method concerns a model based fault de-
tection and isolation used to determine any lock of mo-
tion from an actuated robot joint after contact with static
obstacles. Tso et al. [8] worked on a model-based control
scheme for robot manipulators employing a variable
structure control law in which the actuator dynamics is
taken into consideration. Zhu et al. [9] attached an addi-
tional model-based parallel-compensator to the conven-
tional model-based computed torque controller which is
in the form of a serial compensator to enhance the ro-
bustness of robot manipulator control. Qu et al. [10] have
discussed robust control of robots by the computed
torque law with respect to unknown dynamics by judi-
ciously choosing the feedback gains and the estimates of
the nonlinear dynamics. The choices for the constant
gains depend only on the coefficients of a polynomial
bound of the unknown dynamics. Khosla and Kanade [11]
presented the experimental results of the real-time per-
formance of model-based control algorithms. The com-
puted-torque scheme which utilizes the complete dy-
namics model of the manipulator is compared with the
independent joint control scheme which assumes a de-
coupled and linear model of the manipulator dynamics.
Pathak et al. [12] worked on a scheme for robust trajec-
tory control of space robot and this work presents a re-
duced modal based controller for trajectory control of
flexible space robot in work space. Berger et al. [13]
carried out the application of bond graph modeling to
robots. Special emphasis is placed on adjusting the exact
bond graph to allow for valid numerical solutions.
This paper presents a model based controller which
requires the information of the base of space robot, link
length, joint angle and evaluation of Jacobian. The con-
troller is based on robust overwhelming control of the
flexible space manipulator. To illustrate the methodology,
an example of a two DOF flexible space robot is consid-
ered. The controller is provided with tip velocity infor-
mation of manipulator. Bond graphs are used to represent
both rigid body and flexible dynamics of the link in a
unified manner. The advantage of the bond graph is that
it is used to model system, in multi energy domain and
various control strategies can be devised using this mod-
eling method. SYMBOLS Shakti [14] software is used for
bond graph modeling and simulation.
2. Modeling of Two Arm Flexible Space
Robot
Let us assume that the space robot has single manipulator
with revolute joints and is in open kinematic chain con-
figuration. Figure 1 shows the schematic sketch of two
arm flexible space robot. In this figure {A} represents the
absolute frame, {V} represents the vehicle frame, {0}
frame is located in space vehicle at the base of the robot,
{1} frame is located in first arm at the base of first arm,
{2} frames is located second joint. The frame {t} locates
the tip of the robot. Let L1 be the length of the first link,
L2 be the length of second link and r is the distance be-
tween the robot base and center of mass (CM) of the
space vehicle. Let the first motor is mounted on base and
it applies a torque 1

on first link. Similarly let the sec-
ond motor is mounted on first link and it applies a torque
2

on second link. The cross section area of links are
assumed to be A. The densities of first and second links
are

1 and

2 respectively. The flexible links are uniform
in cross section with flexural rigidities EI1 and EI2. The
flexible link of space robot has been modeled as
Euler-Bernoulli beam.
3. Bond Graph Modeling
The kinematic analysis of flexible links is performed in
order to draw the bond graph as shown in Figure 2. Fig-
ure 2 also shows the pad sub model. From the kinematic
Figure 1. Schematic representation of two arm flexible
pace robot. s

A. KUMAR ET AL. 114
Figure 2. Complete bond graph model of two arm flexible space robot with reduced model based controller.
relations the different transformer moduli used in bond
graph model are derived. Two type of motion of the links
are considered. First motion is the motion perpendicular
to the link and second motion is the rotational motion of
the links. The velocities in perpendicular to link is re-
solved in X and Y direction.
In Figure 2, MV and IV represents the mass of space
vehicle and inertia of space vehicle. Let Xcm and Ycm be
the co-ordinate of the CM of the space vehicle with re-
spect to absolute frame. The first and second link is di-
vided into four segments of equal length. The segment
angles of first and second link are ψ1, ψ2, ψ3, ψ4 and ω1,
ω2, ω3, ω4 respectively. The segment inertias are at-
tached to “1” junction structure corresponding to these
velocities. Pads are used for computational simplicity i.e.,
to avoid the differential causality.
The velocity of the first link at junction 2 in X and Y
direction with respect to absolute frame can be evaluated

A. KUMAR ET AL.115
as
 
22
sincos π2
xcm
XX rY

1

  
  (1)
 
22
cossin π2
ycm
YY rY

1

  
  
 (2)
where 11,

  and 2 Y is the velocity of the link in
direction perpendicular to link at junction 2. The velocity
of the first link tip in X and Y direction with respect to
absolute frame can be evaluated as


_1514 4
8sin
tx
XXL

 
  (3)

_1514 4
8cos
ty
YY L

 
  (4)
where 5
x
X
 and 5y
Y are the velocity of the first link at
junction 5 in X and Y direction.

The velocity of the second link at junction 6 in X and
Y direction with respect to absolute frame can be evalu-
ated as

6_16
cos π2
xt
XXY

1

 
 (5)


6_16 1
sin π2
yt
YYY

 
 (6)
where 6 is the velocity of second link at junction 6 in
the direction perpendicular to link. The velocity of the
robot at tip in X and Y direction with respect to absolute
frame can be evaluated as
Y


_2924 4
8sin
tx
XXL

 
  (7)

_2924 4
8cos
ty
YYL

 
  (8)
where 9
x
X
 and 9y
Y are velocity of the link in X and Y
direction at junction 9.

From the above kinematic analysis different trans-
former modulli used in drawing bond graph can be
evaluated and are shown in Table 1.
4. Reduced Model-Based Controller Design
In work space control the robot is required to track a
given end-effector trajectory, this is achieved by using a
reduced model based controller. The bond graph model
of a two DOF flexible planar space robot controller is
shown in Figure 2. The model is represented in the iner-
tial frame. Controller consists of three parts.
1) Space robot virtual base model
2) Overwhelming controller [12]
3) Jacobian.
4.1. Space Robot Virtual Base Model
A virtual base of the space vehicle is considered. To de-
sign the model based controller, drawing analogy from
the space vehicle, the tip velocity of manipulator in X
and Y direction is considered in controller design. It is
Table 1. Transformer moduli for finding velocities at vari-
ous points in bond graph model.
Description X Direction Y Direction
Robot Base TF = –r sin(  ) TF = r cos(  )
First Segment of
First Link TF = cos(/2 + ψ1) TF = sin(/2 + ψ1)
Second Segment
of First Link TF = cos(/2 + ψ2) TF = sin(/2 + ψ2)
Third Segment
of First Link TF = cos(/2 + ψ3) TF = sin(/2 + ψ3)
Fourth Segment
of First Link TF = cos(/2 + ψ4) TF = sin(/2 + ψ4)
First Link Tip TF = – (L1/8) sin(ψ4) TF = (L1/8) cos(ψ4)
First Segment of
Second Link TF = cos(/2 + ω1) TF = sin(/2 + ω1)
Second Segment
of Second Link TF = cos(/2 + ω2) TF = sin(/2 + ω2)
Third Segment
of Second Link TF = cos(/2 + ω3) TF = sin(/2 + ω3)
Fourth Segment
of Second Link TF = cos(/2 + ω4) TF = sin(/2 + ω4)
Second Link TipTF = – (L2/8) sin(ω4) TF = (L2/8) cos(ω4)
determined by translational inertia Mf of space vehicle.
The displacement relation of the robot are given as


1
212
cos cos
cos
tipf fff
f
XXrL
L
1



 
 (9)


1
212
sin sin
sin
tipf fff
f
YYr L
L
1



 
 (10)
Here
f

is the rotation of the space vehicle.
The tip velocity of the robot tipf
X
, and due to
motion of the space vehicle can be found as,
tipf
Y



 

11
212
11212
2122
sin sin
sin
sin sin
sin
tipf fff
ff
ff
f
XXr L
L
LL
L


1




 

















(11)


 

11
212
11212
2122
cos cos
cos
cos cos
cos
tipf fff
ff
ff
f
YYr L
L
LL
L


1




 

















(12)
If it is assumed that the controller overwhelms the ro-

A. KUMAR ET AL. 116
bot dynamics, then neglecting the coefficients of 1

 and
2

 in Equations (11) and (12) we get,
modulli involve only link lengths and joint angles of the
manipulator.


1
212
sin sin
sin
tipf fff
ff
XXr L
L
1




 





 (13) 4.2. Overwhelming Controller
The linear overwhelming controller [12] is applied to the
tip of the space robot model. The overwhelming control-
ler is provided with reference velocity and tip velocity
information. The effort signal produced by the controller
is magnified by high gain and then the joint torques are
evaluated with the help of Jacobian. This joint torque
information is fed to different joints.


1
212
cos cos
cos
tipf fff
ff
YYr L
L
1




 





 (14)
Using Equation (13) and (14), the transformer modulli
µ5, and µ6 shown in bond graph of Figure 2, can be writ-
ten as,

51121
sin sinsin
ff f
rL L  

 


2

2



4.3. Evaluation of Jacobian
 
61121
cos coscos
ff f
rL L  

 

If we assume that the arms of manipulator are rigid then,
kinematic relations for the tip displacement tip
X
,
in X and Y directions can be written as,
tip
Y
One can observe from these expressions that these




 
1121
11212
cos coscos
sin sinsin
tip CM
tip CM
XXrL L
YYrL L
 
 
  


 
 

 
2

2
(15)
The tip angular displacement with respect to absolute
frame X axis is given as,
The Jacobian of the forward kinematics can be calcu-
lated in bond graph. For the planar case discussed here
this relation can be worked out directly from Equation
(15) as follows,
1tip

   (16)










111 21212
11 1212 12
sin sinsin
cos coscos
tip CM
tip CM
rL L
XX
YYrL L

 
  

 







 

 
 

  (17)
 
 
 
 
112 12
112 12
11212
11212
1 12 2
122 2
sin
cos
sin sin
cos cos
sin sinsin
cos coscos
tip CM
tip CM
XXr
YYr
LL
LL
LL L
LL L
   
 

 
 



 
 


  
 

 

 


 
 















(18)
For the evaluation of joint torque if we assume that
space vehicle rotational velocity

, and CM velocity
CM
X
, CM are small, then neglecting the first and last
term of Equation (18) we get
Y






 
1221
1222
112 12
112 12
sin sinsin
cos coscos
tip
tip
XLL L
YLL L


  
  
 

  





 




 (19)

32 12
sin ,L

   

1
2
tip
tip
XJ
Y










 

 (20)

42 12
cos .L

  
With the help of Equation (20) various transformer
modulli shown in bond graph can be evaluated as
5. Simulation and Results
tation; first and second
int angles are 0 radian, 0.5 radian and 0 radian respec-
 
11 1212
sin sinLL


   
It is assumed that initially base ro
 
211 212
cos cosLL


   jo

A. KUMAR ET AL.117
for the robot tip is a circle of radius R. Then the tip
co
tively. Figure 3 shows the initial configuration of the
space robot. At the beginning of the simulation over-
whelmer initial position is initialized to robot tip posi-
tion.
Let us assume that the reference displacement com-
mand
ordinates will be given as,

0
cos
ref
X
tR tX   (21)



0
sin
ref
Yt RtY   
Here
(22)
00
(,)
X
Y
he corresp
is the center of the
circle. Tonding reference vel
gi,
circular reference
ocity command is
ven by

sin
refx
f
Rt

  (23)

cos
refy
f
Rt

  (24)
At the start of simulation the refe
initialized to tip trajectory. The pa
si
values used for simulation.
rence trajectory is
rameters used for
mulation are shown in Table 2. The simulation is car-
ried out for 1.56 seconds.
Table 2. Parameters and
Parameter Value
Modulus of E E = 70 9m2
lasticity  10 N/
Link Length
Figure 3. Initial configuration of two arm flexible space
robot.
Figure 4(a) shows the plot for reference trajectory in
rection, Figure 4(c) shows the plot for tip trajec-
ry in
this paper we presented a novel reduced model based
ory control of flexible space robot in
Bernoulli beam model is used to
x
direction, Figure 4(b) shows the plot reference trajectory
in y di
to x direction, and Figure 4(d) shows the plot for tip
trajectory in y direction. From Figures 4(a) and (c) it is
seen that tip follows the reference trajectory in x direc-
tion and from Figures 4(b) and (d) it is also seen that tip
follows trajectory in y direction effectively. Due to the
nature of the flexible link tip vibration are observed.
Figure 5(a) shows the plot of base rotation with respect
to time. The base of the space vehicle rotates between the
points 0.0 to –0.03 as the space robot is free floating.
Figure 5(b) shows the plot of first joint angle with re-
spect to time. Figure 5(c) shows the plot of second joint
angle with respect to time Figure 5(d) shows the plot of
CM of space vehicle in x direction with respect to time.
Figure 5(e) shows the plot CM of space vehicle in y di-
rection with respect to time. Figure 5(f) shows the plot
of CM trajectory of space vehicle. Figures 5(d), (e) and
(f) it is seen that there is continues movement in base of
space robot as it is free floating.
Figure 6 shows the tip trajectory error plot. The error
is taken as the difference between reference and actual
tip position. Figure 6(a) shows the error plot for X tip
position with respect to time. Fi
LL
obot Base
CM
m
st A1 = 6.288 –5 m2,
A2 = m2
R
Moment of Inertia of Space I
Im1 = 0.001 kg·m2,
Im2 =·m2
MC = 0.0.13
Internal Damping Coeffi-
terial
Input Reference Angular
Density of Aluminum ρ = 2700 kg/m3
1 = 0.5 m; 2 = 0.6 m
–13 4
Link Inertia I1 = 6.609  10 m,
I2 = 6.609  10–13 m4
Location of R
from Vehicle r = 0.1 m
Mass of the First Link 1 = 1.0 kg
Cross Section Area of Fir
and Second Link
 10
6.288  10–5
Joint Resistances R1 = 0.001 Nm/(rad/s),
2 = 0.0001 Nm/(rad/s)
Mass of Base M = 5.0 kg
Vehicle V = 1.0 kg-m2
Gain Value µH = 8.0
Motor Inertias 0.001 kg
Controller 001, KC = 0.241, RC =
cient for MaRint = 0.01 N/(m/s)
Reference Circle Radius R = 0.400 m
Velocity
 = 1.0 rad/s
gure 6(b) shows the er-
ror plot for Y tip position with respect to time. From the
simulation we see that initially error increases with time
but after some time it start reducing.
6. Conclusions
In
controller for traject
ork space. Eulerw
model flexible link. Reduced model based controller
consists of three parts 1) the model of the virtual base of
space vehicle 2) an overwhelming controller 3) a Jaco-
bian evaluation to evaluate joint torques. The advantage

A. KUMAR ET AL. 118
Time ( s)
X
ref
(m)
Time ( s)
Y
ref
(m)
(a) (b)
Time (s)
X
tip
(m)
Time (s)
Y
tip
(m)
(c) (d)
Figure 4. (a) Reference trajectory in x direction; (b) Referrajectory in y direction; (c) Tip trajectory in x directiond)
Tip trajectory in y direction.
ence t. (
-
-
-
-
-
-
Time (s)
Time (s)
(a) (b)

A. KUMAR ET AL.119
-
-
-
-
Time (s) Time (s)
X
cm (m)
(c)d) (
-
-
-
-
-
-
-
Time ( s)
Y
cm
(m)
-
-
-
-
-
-
-
Y
cm
(m)
X
cm
(m)
(e) (f)
Figure 5: (a) Plot of base rotation versus time; (b) Plot of first joint angle versus time; (c) Plot of second joint angle versus
time; (d) Centre of mass trajectory in x direction; (e) Centre of mass trajectory in y direction; (f) Centre of mass trajectory of
space vehicle.
-
(a) (b)
Figure 6. (a) Error in X tip trajectory with respect to time; (b) Error in Y tip trajectory with respect to time.

A. KUMAR ET AL. 120
of the reduced model based controller is that it does not
require the link dynamic parameters and the trajectory
tracking is very good. However the model has limitation
as compared to analytical model as it is finite approxi-
mation of continuous system. The future work will be
focus on developing a reduced model based controller
for three dimensional flexible space manipulator.
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