G0311047057 by research

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.

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)

Nonlinear Adaptive Control For Welding… where



VE  x E



frame,   

y E





E



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

2 1

L1 YP

3

x xE

p

P  J1

p b

X O

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  X P   X E      , VE   YE  VP   YP  , WP        E  P cos  P  sin  P 0 Rot1   sin  P cos  P  0 0

 0   xE    0  , 1 pE   yE  ,   E   P

0 0 , 1

 E  1   2   3   P 

 2

          ,  E 1 2 3 P

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.

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)

YR Y



YE E

e1 XR

XE y



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       , , 0  p R  YR   p E   YE   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

 e1   E e2  E  R cos e3    e      e   sin e  E E E 1 R 3   2  e3    R  E

where p R  VR and p E  VE . Substituting (1), (3) and (6) into (7) yields E E  A A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 V0  E ET E E To achieve the negative of V , (11) must satisfy V0 

(9) (10)

Nonlinear Adaptive Control For Welding… E E   K E E

(11)

where K  diagk1 k2 k3  with positive values k1, k2 and k3. Substituting (11) into (8) yields KE E   A A 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 ( A A 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 S  e k  e  C  v C  1  R 23e3 2 2 1 E 23 P 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

~ (14) L  L  Lˆ T ˆ where L  L1 L2 L3  , and L is the estimated value of L. Now, (1) becomes VˆE  Jˆd



where d  1d 2 d 3d



(15)

with id is the desired value of i , Jˆ is estimated matrix of J as follows:





~ Jˆ  f  i , Lˆi  J  J

(16)

where

 L~3 S123  L~2 S12  L~1S1  L~3 S123  L~2 S12  L~3 S123   ~ ~ ~ ~ ~ ~ ~ J   L3C123  L2C12  L1C1 L3C123  L2C12 L3C123  (17)  0 0 0   Substituting (16) into (15) yields ~ VˆE  ( J  J )d (18) If Li is estimated, the position vector of the end-effector becomes 1

pˆ E 1p E 1~ pE

 L~1 cos1   L~2 cos1  2   L~3 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

Nonlinear Adaptive Control For Welding… E E  A A 1 E E  A(V R  (V P  W P  0 Rot1 1 pˆ E  0 Rot1 Jˆd )) Substituting (18) and (19) into (21) yields

(21)

E E  A A 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 ~ p E  0 Rot1 J  d ~ AW 0 Rot 1 ~ p and A0 Rot J  can be rewritten as follows





~ AW P  0 Rot1 1 ~ p E  AP L AP     ~  cos 2   3   cos 3   1  L1  ~    P  sin  2   3  sin  3  0   L2  ~  0 0 0   L3  ~ ~ A0Rot1J  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  A A 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

~ ~  ~ V1  EET E E  LT L  EET E E  LˆT L where   diag 1  2  3  with

(26) (27)

 i is the positive definite value and L~T

   LˆT .

Substituting (25) into (27) yields

~  ~ V1  V0  EET ( AP  AJ ) L  LˆT L

(28)

To achieve the negative value of V1 , the control law is chosen as (13), and the parameter update rule is chosen as

 LˆT  EET  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

e6 T is defined as

Nonlinear Adaptive Control For Welding…

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  2   4  2 4

X M  v P cos  P  D P cos  P , Y  v sin   D sin  , M

   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

e4  v E cos e6   1 e5  D  v  e    v sin e    0  e4   P  (33) 6   5  E P   e6    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 V0  e4 e4  e5e5  e6 k5 V0  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)

Nonlinear Adaptive Control For Welding… y

e5 E(XE,YE)

E

M(XM,YM)

e4 Initial configuration of the manipulator

x vP

P P

P(XP,YP)

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 e4  v E cos e6   1 e5  D   r  e    v sin e    0  e4   2 6   5  E r e6    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 e4  v E cos e6   1 e5  D   2a1 e    v sin e    0   e4   6   5  E  1 e6    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  rw  aˆ1     ˆ  lw  a1

aˆ 2   v Pd     aˆ 2   Pd 

(40)

(41)

where v Pd and Pd are the desired values of v P and  P , aˆ1 

1 bˆ and aˆ 2  are the estimated values of a1 and rˆ rˆ

a2. Substituting (41) into (39) yields  aˆ e4  v E cos e6   1 e5  Dˆ   1   a e    v sin e   0  e4   1 6   5  E  e6    E   0 1   0  

 0 v   Pd (42) aˆ 2   Pd  a 2 

 3      3       3  Dˆ  Lˆ1 sin    Lˆ2 sin        Lˆ3 sin         4   4  2   4  2 4 a rˆ rˆ b  2  P  a2 p  p ˆ aˆ 2 b bˆ r

where

 Pd

 Pd 

a1 rˆ v P  rˆa1 P  v P aˆ1 r

Nonlinear Adaptive Control For Welding… Let us define the estimation errors as follows

a~1  a1  aˆ1 , a~2  a2  aˆ 2 Equation (42) can be rewritten as follows

(43)

  a~1  e4  v E cos e6   1 e5  Dˆ   v Pd     v  a e    v sin e    0   (44)  e4    Pd    ~ 1 6   5  E   a 2   e6    E   0  1    Pd    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

V3  e4 e4  e5e5  e6 e6 

 P1a1

a~1a~1 

 P 2 a2

a~2 a~ 2

 a~  1   a~2  1   V2  1  e4 v Pd  aˆ1    e6 Pd  aˆ 2  e4 Dˆ  Pd  (46) a1   P1  a2   P2  ~ ~     where a1   aˆ1 and a 2   aˆ 2 . The controller is still (36) and (37), but there is two update laws as follows

aˆ1   P1e4 v Pd



aˆ 2   P 2 Pd e6  e4 Dˆ

(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

deg.

1.7

0.275 0.395

m m

1 (t=0) 2 (t=0)

135 -90

deg. deg.

3 (t=0)

deg. rad/m

-15 deg 0.007 m/s 5 1.6 /s

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.

Nonlinear Adaptive Control For Welding… Table 2 The parameter values for simulation

bˆ (t=0)

Value Units s 0.11 m

rˆ (t=0)

0.027

2

Lˆ1 (t=0)

0.195

3

Lˆ 2 (t=0) Lˆ (t=0)

0.195

P1

0.195

P2

Parameters

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

10 12 Time (s)

Fig. 6 Tracking error e1 from the simulation and experimental results

Nonlinear Adaptive Control For Weldingâ&#x20AC;Ś 7 6

Error e2 (mm)

5 4 Simulation Result

3 2

Experimental Result

1 0 -1

10 12 Time (s)

Fig. 7 Tracking error e2 from the simulation and experimental results 16 14

Error e3 (deg)

12 10 8 6 Simulation Result

Experimental Result

2 0 -2

10 12 Time (s)

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

10 12 Time (s)

Fig. 9 Tracking error e4 from the simulation and experimental results 4.5 4 3.5 Error e5 (mm)

Simulation Resukt

2.5 2 Experimental Result

1.5 1 0.5 0 -0.5

10 12 Time (s)

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

10 12 Time (s)

Fig. 11 Tracking error e6 from the simulation and experimental results

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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