Advanced Strategies for Robot Manipulators edited by Seyed Ehsan Shafi ei SC I YO Advanced Strategies for Robot Manipulators Edited by Seyed Ehsan Shafi ei Published by Sciyo Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2010 Sciyo All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by Sciyo, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Jelena Marusic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright higyou, 2010. Used under license from Shutterstock.com First published September 2010 Printed in India A free online edition of this book is available at www.sciyo.com Additional hard copies can be obtained from publication@sciyo.com Advanced Strategies for Robot Manipulators, Edited by Seyed Ehsan Shafi ei p. cm. ISBN 978-953-307-099-5 SC I YO.C O M WHERE KNOWLEDGE IS FREE free online editions of Sciyo Books, Journals and Videos can be found at www.sciyo.com [...]... follows: + + ⎧ u1 − x23 / ( r1 J 1 ) ≥ u1 ⎪ u1 = ⎨− x23 ( r1 J 1 ) otherwise − − ⎪ − x23 / ( r1 J 1 ) ≤ u1 u1 ⎩ ( 41) ⎧ − x24 / ( r2 J 2 ) ≥ u u ⎪ u2 = ⎨− x24 ( r2 J 2 ) otherwise − − ⎪ − x24 / ( r2 J 2 ) ≤ u2 u2 ⎩ + 2 + 2 18 Advanced Strategies for Robot Manipulators where the final bound of control for each motor is obtained as: + u 1 = τ 1 − S 1 x 11 − ; u 1 = − τ 1 − S 1 x 11 u = τ 2 − S 2 x 12 − ; u 2... (Usoro, 19 86) Y X 1 (0) = −90 Link - 1 g Link - 2 θ2 (0) = 5 Fig 2 Initial robot configuration 0 .1 FEM C-P P-P Rigid -1. 54 Angular Position-Joint 2 (rad) Angular Position-Joint 1 (rad) -1. 5 -1. 52 -1. 56 -1. 58 -1. 523 -1. 6 -1. 5235 -1. 62 -1. 524 -1. 64 -1. 5245 -1. 66 0 1 2 2 .18 3 2.2 2.22 0.06 0.04 0.02 0 0.085 -0.02 0.084 -0.04 0.083 -0.06 0.082 -0.08 0.0 81 2.24 4 FEM C-P P-P Rigid 0.08 -0 .1 0 5 2.6 2.62 1 2.64... 30 10 20 Torque Motor 2 (N-m) Torque Motor 1 (N-m) 20 30 case 1 case 2 case 3 case 4 upper bound lower bound 0 -10 -20 -30 0 0.5 1 1.5 2 10 0 case 1 case 2 case 3 case 4 upper bound lower bound -10 -20 -30 0 2.5 0.5 1 Time (s) 1. 5 Time (s) Fig 9 Torques of motors Robot Configuration 3 2.5 y(m) 2 1. 5 1 0.5 -1 -0.5 0 0.5 x(m) Fig 10 Robot Configuration 1 1.5 2 2.5 20 Advanced Strategies for Robot Manipulators. .. can be expressed as: x1 (0) = x9 (0) = x10 , x3 (0) = x 11 (0) = X 30 ; (32) x1 ( f ) = x9 ( f ) = X1 f , x3 ( f ) = x 11 ( f ) = X 3 f ; Other boundary conditions are assumed to be zero Now, with defining Z4× 4 = M 4× 4 1 and I 2 × 2 = J 2× 2 1 Eq (30) can be rewritten in the compact form as: q 11 = ( x9 0 0) , (33) q22 = ( x1 x3 ) , q12 = ( x9 x 11 ) , and where q 21 = ( x1 U = ( u1 u2 ) Remember that... ⎡m 11 m12 m13 ⎪⎢ m22 m23 ⎪⎢ ⎪⎢ m33 ⎪ ⎨⎢ Sym ⎪⎣ ⎪⎡ J 0 ⎤ ⎡θ 3 ⎤ ⎡ k1 ⎪⎢ 1 ⎢ ⎥+ 0 J 2 ⎥ ⎣θ 4 ⎦ ⎢ 0 ⎪⎣ ⎦ ⎣ ⎩ m14 ⎤ ⎡θ 1 ⎤ ⎡ h1 ⎤ ⎡ k1 ⎥⎢ ⎥ ⎢ ⎥ ⎢ m24 ⎥ ⎢θ 2 ⎥ ⎢ h2 ⎥ ⎢ 0 + + m34 ⎥ ⎢ e1 ⎥ ⎢ h3 ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ m44 ⎦ ⎢ e2 ⎥ ⎣ h4 ⎦ ⎣ 0 ⎣ ⎦ 0 ⎤ ⎡θ 3 − θ 1 ⎤ ⎡ u1 ⎤ = k2 ⎥ ⎢θ 4 − θ 2 ⎥ ⎢u2 ⎥ ⎦⎣ ⎦ ⎣ ⎦ 0 k2 0 0 0 0 ⎤ ⎡θ 1 − θ 3 ⎤ ⎥⎢ ⎥ 0 0 ⎥ ⎢θ 2 − θ 4 ⎥ =0 0 0⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ 0 0⎦ ⎣ 0 ⎦ (30) 16 Advanced Strategies for. .. the manipulator working performances For that reason, the objective function is formed as: 6 1 2 2 ⎞ L= ⎜ r1 u 1 +r2 u 2 + ∑ w i x 2i ⎟ 2 2⎝ i =1 ⎠ (36) Subsequently, with defining the auxiliary costate vector Ψ = ⎡ψ 1 ψ 2 ψ 12 ⎤ = ⎡ x13 x14 x24 ⎤ results to the Hamiltonian function as: ⎣ ⎦ ⎣ ⎦ H= 6 12 1 2 2 2 ⎞ ⎜ r1u1 +r2 u2 + ∑ w i x 2i ⎟ + ∑ x12 + i xi , 2⎝ i =1 ⎠ i =1 as: (37) Consequently,... up the value of W as: 1, 10 0, and 10 00 Note that in these simulations the penalty matrices refer to velocities of mode shapes are fixed in zero and the payload is assumed to be 1 Kg 1 case 1 case 2 case 3 case 4 0 Angular Velocity-Joint 2 (rad/s) Angular Velocity-Joint 1 (rad/s) 0.5 -0.5 -1 -1. 5 0 0.5 1 1.5 2 case 1 case 2 case 3 case 4 0.5 0 -0.5 -1 -1. 5 -2 0 2.5 0.5 1 Time (s) 1. 5 2 2.5 2 2.5 Time... elements ‘i(j -1) ’ 8 Advanced Strategies for Robot Manipulators ⎡ cos(θ 1 ) − sin(θ 1 )⎤ and ‘ij’ of link i T 01 = ⎢ ⎥ is transformation matrix from body-fixed system ⎣ cos(θ 1 ) sin(θ 1 ) ⎦ attached to link 1 to inertial system of coordinates and 1 is it’s correlated joint angle These energies of elements are then combined to obtain the total kinetic energy T, and potential energy V, for the each link... joints: (a) joint 1; (b) joint 2 2.3 2.35 1 2.4 2.45 2.5 2 3 Time (s) (b) 4 5 10 Advanced Strategies for Robot Manipulators -4 -5 x 10 5 Mode Shape - Link 2 (m) Mode Shape - Link 1 (m) 1 0.5 0 -0.5 P-P C-P -1 0 1 2 3 4 5 Time (s) x 10 0 P-P C-P -5 0 1 2 3 4 5 Time (s) (a) (b) Fig 5 Amplitudes of vibration’s modes: (a) Link 1; (b) Link 2 2.2.2 Dynamic modelling of flexible joint manipulator To model... x3 0 0) , q1 = Z(K (q 21 − q 11 ) − H ) = F1 q 2 = I (U − K (q22 − q12 )) = F2 , x 11 Hence, by defining the vector F as: F = [F1 space equations of system can be written as: x2 i − 1 = x2 i ; x2 i = f i ; F2 ] = [ f 1 i = 1 6 f2 f3 f4 f5 f 6 ] the set of state (34) In order to derive the equations associated with optimality conditions, penalty matrices can be selected as follows: W = diag( w1 , w2 , w3 . presented in (Usoro, 19 86). Link - 1 Link - 2 90)0( 1 −=θ 5)0( 2 =θ g X Y Fig. 2. Initial robot configuration 0 1 2 3 4 5 -1. 66 -1. 64 -1. 62 -1. 6 -1. 58 -1. 56 -1. 54 -1. 52 -1. 5 Time (s) Angular. (a) joint 1; (b) joint 2. Advanced Strategies for Robot Manipulators 10 0 1 2 3 4 5 -1 -0.5 0 0.5 1 x 10 -4 Time (s) Mode Shape - Link 1 (m) P-P C-P (a) 0 1 2 3 4 5 -5 0 5 x 10 -5 Time. Campa Contents Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Real-Time-Position Prediction Algorithm for Under-actuated Robot Manipulator Using