Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
Linearization and Robust Control of Scara Robot Used in
Manufacturing
M. Moavenian 1, M.R. Gharib2, A.A. Akbari 3 and A.H. Samiee 4
Solid Mechanic Engineering Group, Mechanical Engineering Department,
Ferdowsi University, Mashhad, Iran
Corresponding Author E-mail: moaven@um.ac.ir
Abstract
This paper presents a new linearization technique to be used for robust control of an
SCARA robot manipulator. Some nonlinear terms in the model of robot manipulator
are compensated by linear terms which are introduced in the paper. Linearization
process is carried out for four arms but QFT is only applied to design feedback
controller for the first arm. The research indicates that applying the proposed
technique successfully overcomes obstacles for robust control of nonlinear SCARA
robot.
Keywords: SCARA, Robot, Robust Control, Linearization
1 Introduction
SCARA robots are widely used in assembly manufacturing (‘Figure’ 1). The robot is a
horizontally articulated manipulator with a vertical joint at the wrist end. The arm is
very stiff in the vertical direction but is relative compliant laterally. This feature is
convenient for a variety of assembly tasks. The potential problem arises mainly due
to the positioning errors in assembly. Adaptive and model-based controls are two of
the popular control strategies used to control robotic systems. These control
schemes cannot overcome the structure uncertainties of a robotic system. [1]
Dynamic model of robot manipulator consist of highly nonlinear coupled second
order differential equations. Linear time invariant control laws which utilize linear
models of robot manipulator dynamics are often used for industrial robot
manipulators because of the simplicity of the control algorithms. However
nonlinearity and parameter variation in real systems prevents, ordinary linear time
invariant control schemes to achieve satisfactory control performance.
Linearization tecniques for robust control of robot manipulators with uncertainty has
been the subject of many research studies.For example Kawabata, et al. (1993) and
Takayanagi et al. (1993) studied robust position controller for 2 link manipulator. [2]
T.J. Tern, A.K. Bejeay, A. Lotdorl, and Y. Chen have presented dynamic modeling
and linearization technique for a SCARA robot. [3]
1
2
3
4
- Assistant Professor, Mechanic Engineering
- MSc Student, Mechanic Engineering
- Assistant Professor, Mechanic Engineering
- MSc Student, Mechanic Engineering
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
2 Scara robot
2.1
Arm matrix
Arm matrix shows the position and the direction of the tool, with respect to base
coordinate system.
R

tool
T base

0
0

P

0 1 
(1)
Where p is a vector of position and R is a vector of direction for the tool.
The shaping space of the tool has 6 dimensions because every form of tool placing
can be found by its position ( Px , Py , Pz ) and its direction characteristics yaw, pitch,
roll. Generally reversed kinematic problem is as follows:
For each P position and R direction for the tool, find the space variation value to
satisfy the above equation.
If q n shows the roll angel of the tool, then tool forming vector is w that in R 6 is defined
p
W 1  

as follows:W   2   
3 
W   [exp(q n /  )r ] 
For SCARA robot, tool forming vector is as follows.
(2)
W  [ px , py , pz ,0,0, exp( q4 /  )]T
(3)
Figure 1: Selective compliant assembly robot arm (SCARA). [4]
2.2
Inverse kinematic for SCARA robot
‘Figure’ 2 shows the inverse kinematic chart for SCARA robot.[4]
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
Figure 2: Inverse kinematic chart for SCARA robot.
Where si and ci are sin ( qi ) and cos ( qi ) respectively.
2.3
Continuous Movement
If the speed of each joint can be controlled individually, robot can pass continuous
directions. For problems which their inverse kinematic is available analytically, we
can have the speed of each joint calculating the derivative of it with respect to time.
The results as obtained are given respectively as follows: [4]
Figure 3: Link coordinate diagram of the scara robot [4]
q2  
2(w 1w 1  w 2w 2 )
[(2a1a2 )  ((w 1 ) 2  (w 2 ) 2  (a1 ) 2  (a2 ) 2 ) 2 ]1/ 2
2
b1  (a1  a2C 2 )w 1  a2S 2w 2  a2 (S 2w 1  C 2w 2 )q 2
b 2  (a1  a2C 2 )w 2  a2S 2w 1  a2 (C 2w 1  S 2w 2 )q 2
3
(4)
(5)
Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
b1b 2  b 2b1
(b1 )2  (b 2 ) 2
q 3  w 3
w 6
q4 
w6
q1 
(6)
(7)
(8)
2.4
Dynamics of Robot
When the Newton-Euler equations are evaluated symbolically for any manipulator,
they yield a dynamic equation which can be written in the form
D ( q ) q  C ( q, q )  G ( q )  
(9)
Where D(q) is the n  n mass matrix of the manipulator C ( q, q ) is an n  1 vector of
centrifugal and Coriolis terms, and G (q) is an n  1 vector of gravity terms.
2.5
System Linearization
In order to reduce the nonlinear robust controller, design to a much easy and simpler
problem. A linearization technique is presented here, which covers all features of the
real nonlinear system in working space. Each arm is considered as a load system
which is connected to the motor. Then the following simple governing equation can
be considered for each arm.
Ignoring all non linear terms in equation (1), we have J eff   Ceff   
(10)
Angular velocity (  ), acceleration(  ) and required torque (  ) for different trajectories
can be calculated, by simulating the nonlinear model using "Matlab Robotic Toolbox"
based on numerical values found for them, an equivalent linear plant will be derived
for each arm as follows:
 J eff 
   . 
(11)
   
Ceff 
In equation (9), matrix [ ] and vector  can be obtained carrying out identifying
tests [3] or simulating robot in a defined trajectory. J eff and Ceff are identifiable terms
with respect to defined points in every trajectory. Therefore, for n points of a
trajectory we have:
 0

1
.

.

 n
0 
 0 

 
1 
 J eff   1 

. .
 . 
 Ceff   
 .
.  
 

 n 
 n 
 
 0

1
Suppose A   .

.

 n
0 
 0 

 
1 
 1

. , B= . 

 
. 
. 

 n 
 n 
 
(12)
 J eff 
†
†
T
-1
T
We have 
(13)
  A .B . Where A =(A A) A
C
 eff 
For example equation (9) of SCARA robot used in [5] has the following given
numerical values.
4
Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
 .538  .237 cos 2
 .192  .118cos 
2
D

0

085

By
.192  .118cos 2
.195
0
085
0 
.085
1.003 0 

0
.088
0
0
2


  (sin 2 )(.23712  .118  2 ) 


2

C
.118  1 sin 2


0




0


 0 
 0 

G
 9.87 


 0  running the simulation
(14)
for multiple trajectories of all arms, J eff and
C eff are found respectively.
J eff  [.2 .5] Ceff  [.75
5.2] Arm 1
(15)
J eff  [.14 .26] Ceff  [.22 2.12] Arm 2
(16)
J eff  [.003 .006] Ceff  [.01 .4] Arm 3
(17)
J eff  [.15 .25] Ceff  [.52 .84] Arm 4
(18)
The above founded values were chosen, selecting certain known operating points
which sufficiently represent the range of variation in joint dynamics.
As a result linearized, transfer function for each arm can be represented as follows:
1
i=1,2,3,4
(19)
Pi 
s( J eff .s  Ceff )
3 QFT controller design
Using QFT method ([6], [7], [8], [9]) for controlling the selected SCARA. The
nonlinear plant needs to be converted to family of linear and uncertain processes
implementing the new technique mentioned in part 2.5.
A suitable QFT controller and prefilter ‘Figure’4 were then designed for the four joints
to satisfy the closed loop specifications (( Mp =20%) and ( Ts =0.08 s) where Mp and
Ts are the overshoot and the settling time respectively).
Figure 4: Structure of a two degrees of freedom system
Template generation (reveals frequency domains ‘Figure’ 5); robust margins for five
selected trajectories of the first arm based on frequencies found in template
generation are shown in ‘Figure’ 6, robust tracking and bounds intersection of the
same arm are presented in ‘Figures’ 7 and 8 respectively. [6]
5
Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
Plant Templates (parametric part w/o hardware)
-45
30
-50
Magnitude (dB)
-55
50
-60
80
-65
100
-70
140
-75
-80
-350
-300
-250
-200
-150
Phase (degrees)
-100
-50
0
Figure 6: Robust margin for arm 1
Figure 5: Template generation
Figure 8: Intersection of bounds for
arm1
Figure 7: Robust tracking for arm 1
The loop shaping and prefilter functions are found running QFT toolbox. The results
are presented in ‘Figures’ 9 and 10 respectively. [6], [9]
Figure 9: Loop-shaping in Nichols chart for arm 1
The respected controller is found as:
G1 ( s )  182.65(102 
200
 s)
s
(20)
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
Figure 10: Pre-filter shaping in Nichols chart for arm 1
The respected prefilter is found as:
F1 ( s ) 
94080
s  108.6 s  5088 s  94080
3
(21)
2
The time domain closed loop response before and after adding the designed
controller and prefilter are shown respectively in ‘Figures’ 11 and 12 which indicate
the strength and practibility of the proposed design method and supports the
frequency domain results.[6]
1.2
1
Y(t)
0.8
0.6
Output without Controller & prefilter
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
0.4
Time(s)
0.5
0.6
0.7
0.8
Figure 11: Time domain simulation without controller and prefilter for arm 1
Time Domain Simulation
Upper Bound
1.2
1
Y(t)
0.8
0.6
Desined Controller Output
0.4
Lower Bound
0.2
0
-0.2
0
0.01
0.02
0.03
0.04
0.05
Time(s)
0.06
0.07
0.08
0.09
0.1
Figure 12: Time domain simulation for arm 1
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Tehran International Congress on Manufacturing Engineering (TICME2007)
December 10-13, 2007, Tehran, Iran
4 Conclusion
This paper proposes new linearization technique implemented for controlling an
SCARRA using QFT method. The technique is not restricted to QFT and can be
tested on other methods as well. Our research indicates that applying the proposed
technique successfully overcomes obstacles for robust control of nonlinear SCARA
robot. A comparative verification of the technique will show the strength of our
findings.
References
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[2] Z. B. Kong, T. Y. Chai, K. Oshima, J. M. Yang, and S. Fujii, “Robust Vibration
Control of Scara-Type Robot Manipulators“. http://www.paper.edu.cn.
[3] T.J. Tern, A.K. Bejeay, A. Lotdorl, and Y. Chen “Nonlinear Feedback in Robust
Arm Control“, Proce, IEEE control, NY, Dec. (1984) pp. 38-51.
[4] M.S. Asnaashari,“Design and product of a software for simulating behavior of
robot manipulators in inverse kinamatics and control problems by classic and
neuro-fuzzy methods “ Master of Science Thesis, Sept. (2005).
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ICEE Conference, Isfahan, (2000) pp. 161-168.
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Symposium on Automation and Robotics in Construction (ISARC), Kochi, Kerala,
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Vol. 33, (1991) pp. 255-261.
[8] A.C. Zoloas, and G.D. Halikias, “Optimal Design of PID Controllers Using the
QFT Method“, IEE Proc-Control Theory Appl, Vol. 146, (1999)
[9] P.S.V. Nataraj, “Computation of QFT Bounds for Robust Tracking Specifications“,
Automatica 38, (2002) pp. 327-334.
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