The document presents an adaptive tracking control method for a welding mobile manipulator. The mobile manipulator consists of a three-linked planar manipulator mounted on a two-wheeled mobile platform. Controllers are designed using Lyapunov stability analysis to guarantee the end-effector tracks a desired welding trajectory despite unknown dimensional parameters of the manipulator. Simulation and experimental results demonstrate the effectiveness of the proposed adaptive controllers.

1. Research Inventy: International Journal Of Engineering And Science
Vol.3, Issue 11 (November 2013), PP 47-57
Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com
Nonlinear Adaptive Control for Welding Mobile Manipulator
1,
Ho Dac Loc, 2,Nguyen Thanh Phuong, 3,Ho Van Hien
Hutech High Technology Instituite
ABSTRACT : This paper presents an adaptive tracking control method for a welding mobile manipulator with
a kinematic model in which several unknown dimensional parameters exist. The mobile manipulator consists of
the manipulator mounts on a mobile-platform. Based on the Lyapunov function, controllers are designed to
guarantee stability of the whole system when the end-effector of the manipulator performs a welding task. The
update laws are also designed to estimate the unknown dimensional parameters. The simulation and
experimental results are presented to show the effectiveness of the proposed controllers.
I.
INTRODUCTION
Nowadays, the welding robots become widely used in the tasks which are harmful and dangerous for
the workers. The three-linked planar manipulator with a torch mounted at the end-effector has three degree-offreedom (DOF). Hence it should be satisfies for three constraints of welding task such as two direction
constraints and one heading angle constraint. However, because a fixed manipulator has a small work space, a
mobile manipulator is used for increasing the work space by placing the manipulator on the two-wheeled
mobile-platform. The mobile manipulator has five DOF such as three DOF of manipulator and two DOF of
mobile-platform, whereas the welding task requires only three DOF. Hence, the mobile manipulator is a
kinematic redundant system. To solve this problem, two constraints of linear and rotational velocity constraints
of mobile-platform are combined into the mobile manipulator system. This means that the mobile-platform
motion is not a free motion but it is a constrained motion. The target of two constraints is to move the
configuration of manipulator into its initial configuration to avoid the singularity.
In recent years, the analysis and control of the robots are studied by many researchers. Most of them
developed the control algorithm using the dynamic model or combining the dynamic model with the kinematic
model. Fierro and Lewis [5] developed a combined kinematic/torque control law of a wheeled mobile robot by
using a back-stepping approach and asymptotic stability is guaranteed by Lyapunov theory. Yamamoto and Yun
[6] developed a control algorithm for the mobile manipulator so that the manipulator is always positioned at the
preferred configurations measured by its manipulability to avoid the singularity. Seraji [2] developed a simple
on-line coordinated control of mobile manipulator, and the redundancy problem is solved by introducing a set of
user-specified additional task during the end-effector motion. He defined a scalar cost function and minimized it
to have non-singularity. Yoo, Kim and Na [3] developed a control algorithm for the mobile manipulator based
on the Lagrange’s equations of motion. Jeon, Kam, Park and Kim [7] applied the two-wheeled mobile robot
with a torch slider welding automation. They proposed a seam-tracking and motion control of the welding
mobile robot for lattice-type welding. Fukao, Nakagawa and Adachi [4] developed an adaptive tracking
controller for the kinematic/dynamic model with the existing of unknown parameters, but the algorithm in their
paper is applied only to the mobile robots. Phan, et al [8] proposed a decentralized control method for five DOF
mobile – manipulator with exactly known with parameters. These methods reveal that the adaptive control
method for the kinematic model of the mobile manipulator has studied insufficiently. For this reason, an
adaptive control algorithm for the kinematic model of the mobile manipulator is the study target of this paper. It
is assumed that all dimensional parameters of the mobile manipulator are unknown and the controllers estimate
them by using the update laws. Finally, the simulation and the experimental results are presented to show the
effectiveness of the proposed method.
II.
SYSTEM MODELING
A Kinematic Equations of The Manipulator
Consider a three-linked manipulator as shown in Fig. 1. A Cartesian coordinate frame is attached at the
joint 1 of the manipulator. Because this frame is fixed at the center point of mobile-platform and moves in the
world frame (Frame X_Y), this frame is called the moving frame (Frame x_y).
The velocity vector of the end-effector with respect to the moving frame is given by

VE  J .
1
(1)
47

2. Nonlinear Adaptive Control For Welding…
where


VE  x E
1

frame,   
1

yE

2

3


E
T

T
is the velocity vector of the end-effector with respect to the moving
is the angular velocity vector of the joints of the manipulator, and J is the Jacobian
matrix.
 L3 S123  L2 S12  L1 S1
J   L3C123  L2 C12  L1C1


1

 L3 S123  L2 S12
L3 C123  L2 C12
1
 L3 S123 
L3 C123  (2)


1

where L1 , L2 , L3 are the length of links of the manipulator, S1  sin(1 ) , S12  sin(1  2 ) , S123  sin(1  2  31) ,
C1  cos(1 ) , C12  cos(1  2 ) C123  cos(1  2  31) .
Y
YE
E
y
yE
J3
L2
2
1
J2
L1
YP
vE
3
L3
vP
x
xE
p
P  J1
p
b
X
O
Xp
XE
Fig. 1
Scheme for deriving the kinematic equations of the mobile manipulator
The end-effector’s velocity vector with respect to the world frame from the motion equation of a rigid body in a
plane is obtained as follows
VE  VP  WP  0 Rot1 1 p E  0Rot1 1VE
(3)
where:
VE : the velocity vector of the end-effector with respect to the world frame
VP : the velocity vector of the center point of platform with respect to the world frame
WP : the rotational velocity vector of the moving frame
0
Rot1 : the rotation transform matrix from the moving frame to the world frame
1
pE : the position vector of the end-effector with respect to the moving frame


XP
XE 
  ,
  ,
VE   YE  VP   YP  WP


 P 
 E 
 
 
cos  P  sin  P
0
Rot1   sin  P cos  P

 0
0

 0 
 xE 
  0  , 1 pE   y E  ,
 
 

E 
 P 
 
 
0
0 ,

1

 E  1   2   3   P 

2
 



,  E  1   2  3   P
B
Kinematic Equations of the Mobile Platform
When the mobile-platform moves in a horizontal plane, it obtains the linear velocity vp and the angular
velocity p. The relation between vp; p and the angular velocities of two driving wheels is given by
 rw  1 r b r   v P 
   
 .
 lw  1 r  b r   P 
(4)
where  rw , lw are the angular velocities of the right and left wheels.
48

3. Nonlinear Adaptive Control For Welding…
CONTROLLERS DESIGN
III.
A. Controller Design For Manipulator
It is assumed that the dimensional parameters of L1, L2 and L3 are known exactly. The coordinate of the
mobile manipulator with the reference welding trajectory is shown in Fig. 2. Our objective is to design a
controller so that the end-effector with the coordinates E( X E , YE ,  E ) tracks to the reference point
R X R YR
 R  . The tracking error vector E E  e1 e2
 e1   cos E
E E  e2    sin  E
  
e3   0
  
sin  E
0
0

1

cos E
0
e3  T is defined as follows
X R  X E 
 Y  Y  . (5)
E 
 R
 R   E 


(5)
R
YR
Y
vR
YE

R
YR
e2
E
YE
E
vE
E
e3
e1
XR
XE
y
x

P
X
XE XR
Fig. 2
Scheme for deriving the error equations of the manipulator
Let us denote
 cos  E
A   sin  E

 0

sin  E
cos  E
0
0
X R 
X E 
,
Y  , p  Y 
0 pR  R
E
 
 E
 R 
 E 
1

 
 
Equation (5) can be re-expressed as follows
EE  A pR  pE  .
(6)


E E  A p R  p E   AVR  VE 
(7)
and its derivative is

 e1   E e2  E  R cos e3 


E E  e2      E e1   R sin e3 
  

e3  

R  E
  



where p R  VR and p E  VE . Substituting (1), (3) and (6) into (7) yields



EE  AA1EE  A(VR  (VP  WP 0 Rot1 1 p E  0Rot1 J )) (8)
The chosen Lyapunov function and its derivative are
1 T
EE EE
2
T 

V0  E E E E

To achieve the negative of V , (11) must satisfy
V0 
(9)
(10)
0
49

4. Nonlinear Adaptive Control For Welding…

EE  K EE
(11)
where K  diagk1 k2 k3  with positive values k1, k2 and k3.
Substituting (11) into (8) yields


KE E   AA 1 E E  A(VR  (VP  WP  0 Rot1 1 p E  0Rot1 J )) (12)
If VP is chosen in advanced as a constraint, the errors converge to zero if the non-adaptive control law of the
manipulator is given as follows



  J 1 0 Rot1 1 ( A 1 ( AA 1  K ) E E  VR  VP  WP 0 Rot1 1 p E ) (13)
(13) is the controller of the manipulator, and it can be re-expressed as follows
 v R S 3e3  v P C12  e2 k 2  e1 E C3



  e k  e   L   e k   S    P
1 1
2 E
3
R
3 3
E
3

  v R L1S 23e3  L2 S3e3  v P L1C1  L2C12 


1 
   E e1  e2 k 2 L1C23  L2C3 

2 
L1L2 S 2 
  L1S12  L2 S3 L3e3 k3   E   L3 R  e1k1  e2 E 



v R S 23e3  e 2 k 2  e1 E C 23  v P C1

1 



3 
L2 S 2   e 2 E  k1e1  L3  R  e3 k 3   E S 23 


  R  e3 k 3   E 
where S 23e3  sin ( 2   3  e3 ) and S 3e3  sin( 3  e3 ) .

1 
1
L1S 2


When the length of link Li is known not exactly, an adaptive controller is designed to attain the control objective
by using the estimated value of Li.
Let us define the estimation error vector as follows
~
ˆ
(14)
L  LL
T
ˆ is the estimated value of L. Now, (1) becomes
where L  L1 L2 L3  , and L
ˆ
ˆ
VE  J d
1


  
where  d  1d  2 d  3d

T
(15)
 ˆ
with id is the desired value of  i , J is estimated matrix of J as follows:


~
ˆ
ˆ
J  f  i , Li  J  J
(16)
where
~
~
~
~
~
~
 L3 S123  L2 S12  L1S1  L3 S123  L2 S12  L3 S123 

~ ~
~
~
~
~
~
J   L3C123  L2C12  L1C1 L3C123  L2C12 L3C123  (17)

0
0
0 


Substituting (16) into (15) yields
~ 
ˆ
VE  ( J  J ) d
(18)
If Li is estimated, the position vector of the end-effector becomes
1
1
ˆ
p E 1p E 1~ E
p
~
~
~
 L1 cos1   L2 cos1  2   L3 cos1  2  3 
~

~
~
1~
pE   L1 sin 1   L2 sin 1  2   L3 sin 1  2  3  


0


(19)
(20)
Now, (8) can be rewritten as
50

5. Nonlinear Adaptive Control For Welding…


E E  AA 1 E E
ˆ
ˆ
 A(V R  (V P  W P  0 Rot1 1 p E  0 Rot1 J d ))
Substituting (18) and (19) into (21) yields
(21)


E E  AA 1 E E  A(V R  (V P  W P  0 Rot1 1 p E
(22)
~

 0 Rot1 J d ))  A W P x 0 Rot1 1 ~ E  0 Rot1 J  d
p
~
AW 0 Rot 1 ~ and A0 Rot J  can be rewritten as follows
p

P
1
E
1

d
~
AW P  0 Rot1 1 ~ E  AP L
p

AP

~
 cos 2   3   cos 3   1  L1 
~ 
  P  sin  2   3 
sin  3 
0   L2 

 ~

0
0
0   L3 

 
~
~
0
A Rot1J  d  AJ L
(23)
AJ 


~
 1 cos 2  3   1  2 cos3   1  2  3   L1  (24)

 ~ 
  1 sin  2  3 
1  2 sin 3 
0
  L2 
~

 L 
0
0
0

  3
Equation (22) can be re-expressed as follows








E E  AA 1 E E  A(V R  (V P  W P  0 Rot1 1 p E
(25)
~

 0 Rot1 J d ))   AP  AJ L
The chosen Lyapunov function and its derivative are
V1 
1 T
1~ ~
E E E E  LT L
2
2
~ ~
 ~

T 
T 

ˆ
V1  EE EE  LT L  EE EE  LT L
where   diag 1  2  3  with
(26)
(27)
~

 i is the positive definite value and LT

ˆ
  LT .
Substituting (25) into (27) yields
~ ˆ ~

T
 
V1  V0  EE ( AP  AJ ) L  LT L
(28)

To achieve the negative value of V1 , the control law is chosen as (13), and the parameter update rule is chosen
as

T
ˆ
LT  EE  AP  AJ  1
(29)
a. Controller Design for Mobile Platform
The task of the mobile latform is to move so as to avoid the singularity of the manipulator’s configuration. A
simple algorithm is proposed for the mobile-platform to avoid the singularity by keeping its initial configuration
throughout the welding process.
The initial configuration of the manipulator is chosen as shown in Fig. 3. Let us define a point M  X M , YM  that
coincides to the point E of the end-effector at beginning. If M and E do not coincide, the errors exist. The error
vector E M  e4
e5
e6 T is defined as
51

6. Nonlinear Adaptive Control For Welding…
EM
e4   cos  M
 e5    sin  M
  
e6  
0
  
sin  M
0  X E  X M 
0  YE  YM  (30)


1   E   M 


cos  M
0
In order to keep the configuration of the manipulator away from the singularity, the mobile-platform has to move
so that the point M tracks to the point E. Consequently, the initial configuration of the manipulator is maintained
throughout the welding process, and the singularity does not appear. The initial configuration of the manipulator
3


,  2   ,  3  . From Fig. 4, we get the geometric relations as
is chosen as 1 
4
2
4
X M  X P  D sin  P ,
YM  YP  D cos  P ,
M  P
(31)
 3    
 3     
 3 
where D  L1 sin    L2 sin        L3 sin        Hence, we have
 4

 4

 4 
 2 
 2 4



X M  v P cos  P  D P cos  P ,

Y  v sin   D sin  ,
M
P
P
P
P

 M  P
(32)
A controller for the mobile-platform will be designed to achieve ei  0 when t .
The derivative of (30) can be re-expressed as follows

e4  v E cos e6   1 e5  D 
v 
e    v sin e    0

 e4   P 
(33)
6 
 5  E
 
 P
e6    E   0
1 
  
 

The chosen Lyapunov function and its derivative are
V0 
1 2 1 2 1  cos e6
e4  e5 
2
2
k5
(34)
sin e6




V0  e4 e4  e5e5 
e6
k5

V0  e4 (v P  D P  v E cos e6 )
(35)
sin e6
( P   E  k 5 e5 v E )
k5

An obvious way to achieve negativeness of V 2 is to choose (vP ,  P ) as

vP  DP  vE cos e6  k4e4
P  E  k5e5vE  k6 sin e6
where k 4 , k5 , and k6 are positive values.
(36)
(37)
52

7. Nonlinear Adaptive Control For Welding…
y
Y
e5 E(XE,YE)
E
vE
M(XM,YM)
e4
Initial configuration
of the manipulator
D
x
vP
e6
P
P
P(XP,YP)
X
Fig. 3
Scheme for deriving the error equations of mobile-platform
When the wheel radius r and the distance from the center point to the wheel b are unknown, we design an
adaptive controller by using the estimated values of r and b.
Substituting (3) into (33) yields

e4  v E cos e6   1 e5  D   r

e    v sin e    0

 e4   2
6 
 5  E
 r
e6    E   0
 1   2b
  
 

Let us define a1 
r 
2  rw  (38)

r 
  lw 
2b 
1
b
and a 2  . Equation (38) becomes
r
r
1

e4  v E cos e6   1 e5  D  
e    v sin e    0
  2a1

 e4  
6 
 5  E
 1
e6    E   0
1  
  
 
  2a 2
1 
2a1   rw 

1  lw 



2a 2 

(39)
Equation (4) becomes
 rw  a1 a 2   v P 
   a  a   
2  P
 lw   1
If r and b are unknown, (40) becomes
ˆ
 rw  a1
    ˆ
 lw  a1
ˆ
a2   v Pd 

ˆ 
 a2   Pd 
(40)
(41)
ˆ
where v Pd and Pd are the desired values of v P and  P , a1 
ˆ
1
b
ˆ
and a2  are the estimated values of a1 and
ˆ
ˆ
r
r
a2. Substituting (41) into (39) yields
ˆ
ˆ a

e4  v E cos e6   1 e5  D   1

 a1
e    v sin e   0

 e4  
6 
 5  E

e6    E   0
  
1   0

 


0
v 
 Pd (42)
ˆ
a 2   Pd 


a2 

 3     ˆ
 3     
 3  ˆ
ˆ ˆ
D  L1 sin    L2 sin        L3 sin       
 4

 4

 4 
 2 
 2 4


ˆ
ˆ
a
r
rb
 2  P  a2 p 
p
ˆ
ˆ
ˆ
a2
b
br
where
 Pd
53
 Pd 
ˆ
a1
r
ˆ
v P  ra1 P  v P
ˆ
a1
r
,

8. Nonlinear Adaptive Control For Welding…
Let us define the estimation errors as follows
~
ˆ ~
ˆ
a1  a1  a1 , a2  a2  a2
Equation (42) can be rewritten as follows
(43)
~
 a1

ˆ 

e4  v E cos e6   1 e5  D 
 a v Pd  
 v 
e    v sin e    0

  (44)
 e4    Pd    ~ 1
6 
 5  E
 
 a2   
e6    E   0
Pd 
 1    Pd  
  
 


 a2

The chosen Lyapunov function and its derivative are given as
V3 
1 2 1 2 1 2
1 ~2
1 ~2
e4  e5  e6 
a1 
a2 (45)
2
2
2
2 P1a1
2 P 2 a2




V3  e4 e4  e5e5  e6 e6 
1
 P1a1
~~

a1a1 
1
 P 2 a2
~~

a2 a2
~
~

1   a2 
1 
 a 
ˆ
ˆ 
ˆ
 V2  1  e4 v Pd 
a1    e6 Pd 
a2  e4 D Pd  (46)



a1 
 P1  a2 
 P2

~
~   a and a   a .




ˆ1
ˆ2
where a1
2
The controller is still (36) and (37), but there is two update laws as follows

ˆ
a1   P1e4 v Pd


ˆ
ˆ
a 2   P 2 Pd e6  e4 D
(47)

(48)
IV.
SIMULATION AND EXPERIMENTAL RESULTS
Table 1 shows the parameter and initial values for the welding wheeled mobile manipulator system used in this
simulation.
Table 1
Numerical values and initial values for simulation
Value
Units
s
XR - XE (t=0) 0.005 m
YR - YE (t=0) 0.005 m
Parameters
ΦR  ΦE
(t=0)
XE (t=0)
YE (t=0)
 E (t=0)
νR
k4
Parameters
k1
k2
Value
Units
s
1.4
/s
1.5
/s
15
deg.
k3
1.7
/s
0.275
0.395
m
m
1 (t=0)
2 (t=0)
135
-90
deg.
deg.
3 (t=0)
45
k5
65
deg.
rad/m
-15 deg
0.007
m/s
5
1.6
/s
k6
2
1.25 rad/s
In this case, the update laws in (29), (47) and (48) are used to estimate the unknown parameters. Parameter
values for simulation are shown in Table 2.
A welding mobile manipulator of prototype as shown in Fig. 4 has constructed to check the effectiveness of the
algorithm.
54

9. Nonlinear Adaptive Control For Welding…
Table 2
The parameter values for simulation
ˆ
b (t=0)
Value
Units
s
0.11 m
ˆ
r (t=0)
0.027
m
2
ˆ
L1 (t=0)
0.195
m
3
ˆ
L2 (t=0)
ˆ
L (t=0)
0.195
m
P1
0.195
m
P2
Parameters
3
Parameters
1
Value
Units
s
0.265
0.187
5
0.163
5
4640
12.4
Fig. 4
The wheeled mobile manipulator prototype
The simulation and experimental results of the system with all exactly unknown parameters are presented as the
following figures 5-11.
Mobile Platform Trajectory
Mobile
Platform
Manipulator
Welding Reference
Trajectory
Fig. 5
The trajectory reference and mobile manipulator posture
6
5
Simulation Result
Error e1 (mm)
4
3
Experimental Result
2
1
0
-1
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Fig. 6
Tracking error e1 from the simulation and experimental results
55

10. Nonlinear Adaptive Control For Welding…
7
6
Error e2 (mm)
5
4
Simulation Result
3
2
Experimental Result
1
0
-1
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Fig. 7
Tracking error e2 from the simulation and experimental results
16
14
Error e3 (deg)
12
10
8
6
Simulation Result
4
Experimental Result
2
0
-2
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Fig. 8
Tracking error e3 from the simulation and experimental results
5
Error e4 (mm)
4
3
Simulation Result
2
Experimental Result
1
0
-1
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Fig. 9
Tracking error e4 from the simulation and experimental results
4.5
4
3.5
Error e5 (mm)
3
Simulation Resukt
2.5
2
Experimental Result
1.5
1
0.5
0
-0.5
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Fig. 10
Tracking error e5 from the simulation and experimental results
8
6
Error e6 (deg)
4
2
Experimental Result
0
-2
-4
Simulation Result
-6
-8
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Fig. 11
Tracking error e6 from the simulation and experimental results
56

11. Nonlinear Adaptive Control For Welding…
V.
CONCLISIONS
This paper proposed an adaptive tracking control method for the mobile platform with the unknown
parameters. Two independent controllers are proposed to control two subsystems. The controllers are based on
the Lyapunov function to guarantee the tracking stability of the welding mobile robot. The dimensional
parameters of the mobile manipulator are considered to be unknown parameters which are estimated by using
update law in adaptive control scheme. The simulation and experiment results show that all of the errors are
converged. The controllers are simple. Those are implemented for microcontroller easily.
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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W. S. Yoo, J. D. Kim, and S. J. Na “a study on a mobile Platform-Manipulator Welding System for Horizontal Fillet Joints”
Mechatronics, Vol. 11, pp. 853-868, 2001.
T. Fukao, H. Nakagawa, and N. Adachi “Adaptive Tracking Control of a Non-holonomic Mobile Robot” Trans. Robo. Auto.,
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57