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# Research Inventy : International Journal of Engineering and Science

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Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.

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### Transcript of "Research Inventy : International Journal of Engineering and Science"

2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] Ph. J. Mc Kerrow Introduction to Robotics, Addison-Wesley Publishers, 1991. H. Seraji “Configuration control of cover-counted canipulators” Proc. of the IEEE Int. Conf. Robo. Auto.,Vol.3, pp. 2261-2266, 1995. 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., Vol. 16, No. 5, pp. 609-615, 2000. Fierro R., and Lewis F. L., “Control of a Non-holonomic Mobile Robot: Backstepping Kinematics into Dynamics”, Proc. IEEE. Conf. Deci. Cont., Vol. 4, pp. 3805-3810, 1995. Yamamoto, Y., and Yun, X., “Coordinating Locomotion and Manipulation of A Mobile Manipulator”, Proc. IEEE Int. Conf. Deci. Cont., pp. 2843~2848, 1992. Jeon, Y. B., Kam, B. O., Park, S. S., and Kim, S. B., “Modeling and Motion Control of Mobile Robot for Lattice Type Welding”, Int. J. (KSME)., Vol. 16, No 1, pp. 83~93, 2002. T. T. Phan, T. L. Chung, M. D. Ngo, H. K. Kim and S. B. Kim, “ Decentralized Control Design for Welding Mobile Manipulator”, Int. J (KSME)., Vol. 19, No 3, pp. 756~767, 2005. 57