From all and a12 the full dynamic model of the system

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friction coefficients, and rml, rm2 are the motor torques. From (All) and (A12), the full dynamic model of the system can be obtained. This is given by D(Q)Q + GQ + H(Q, Q) + #(01, ~)2) = Prm, Q = [O102qlq2" "qee] T, Tm= [Vmlrrn2] T, D(Q) = dl + Jml M12 o~ d2 + Jm2 + Pl5 Z q2 i=l -2a I , _mLlldPl(15) ) ---~- pt 1 x cos(O 2 -- 01) (- 20-~2 PlI -- mLlldP2(15)) ... x Cos(O2 -- 01) M21 ot I +mL(Is +/5)~b1(15) ot2 +mL(ls + 15)02(15) "'" pl 5 + mLq51 (/5) 2 mL~b 1 (/5)~b2(15) m L ~bl (15)~b2 (/5) pl 5 + mLq52 (/5)2 +rnLZ ~i (15)qi i=I ~,4~j (15)qj j=l (-2al _mLlldPl(15)) ~t 1 + 15)~bl (/5) El pll +mL(ls x Cos(O2 -- 01) ( ~Pll -- mLIlq~2(15)) ot2 + mL(ls +15)~2(15) x cos(02 -- O1 )
II II + + + ~ - + + + ~ + + + ~ ~ + + + ~ "~+ + + ~ ~ + I ~ I ~ I ~ + + " - ~M M ~" ' A i'~ ~ ~i'] ~ - .-. + ~ + ~. *~ ~-. ~ ~.. _~ ~/[vj8 - _ ~. "-" = "-" = ~ 8 .~ 8 ~. ~.~ ~- ~. ~ ~. ~_~ _~" II ~. '... ~ 0 ~ I ++ b~b~ I II I I 6
522 Koh Tuck Lye et al d3 = m3121c3 - ½ mslll5 + m411lc4 - m51sll - mLll(ls + 15) + mclllc, M12 = M21 = d3 cos(02 - 01) + 2pll E cr~ flii qi -t- roLl1 Z dPi(15)qi sin(O2 - 01), i=1 i=1 oti = 2p ls ~ + . References Aoustin Yet al 1994 Experimental results for the end-effector control of a single flexible robotic arm. IEEE Trans. Control Syst. Technol. 2:371-381 Banavar R N, Dominic P 1995 An LQG/H-infinity controller for a flexible manipulator. IEEE Trans. Control Syst. Technol. 3:409-416 Book W J 1984 Recursive Langrangian dynamics of flexible manipulor arms. Int. J. Robotics Res. 3:87-101 Cannon R H, Schmitz E J 1984 Initial experiments on the end-point control of a flexible one-link robot. Int. J. Robotics Res. 3:62-75 Craig R R Jr (ed.) 1981 Structural dynamics: An introduction to computer methods (New York: John Wiley & Sons) De Luca A, Siciliano B 1993 Inversion based nonlinear control of robot arms with flexible links. AIAA J. Guidance Control Dyn. 16:1169-1176 Gross E, Tomizuka M 1994 Experimental flexible beam tip tracking with a truncated series approximation to uncancellable inverse dynamics. IEEE Trans. Control Syst. Technol. 2: 382- 391 Hastings G G, Books W J 1986 Verification of a linear dynamic model for flexible robotic manip- ulators. Proe. of the IEEE Conference on Robotics and Automation (Washington, DC: IEEE Comput. Soc. Press) pp 1024-1029 Khorrami F, Jain S, Tzes A 1994 Experiments on rigid body-based controllers with input pre- shaping for a two-link flexible manipulator. IEEE Trans. Robotics Autom. 10:55-64 Lin L C, Yib T W 1996 Rigid model-based neural network control of flexible-link manipulators. IEEE Trans. Robotics Autom. 12:595-601 Matsuno F, Fukushima S 1987 Feedback control of a flexible manipulator with a parallel drive mechanism. Int. J. Robotics Res. 6:76-84 Mills J K, Lokhorst D M 1993 Control of robotic manipulators during general task execution - A discontinuous control approach. Int. J. Robotics Res. 12:146-163 Moudgal V G, Passino K M, Yurkovich S 1994 Rule-based control for a flexible link robot. IEEE Trans. Control Syst. Technol. 2:392-405 Sakawa Y, Matsuno F 1986 Modelling and control of a flexible manipulator with a parallel drive mechanism. Int. J. Control 44:299-313 Siciliano B, Book W J 1988 A singular perturbation approach to control of lightweight flexible manipulators. Int. J. Robotics Res. 7:79-90 Siciliano B, Prasad J V R, Calise A J 1992 Output feedback two-time-scale control of multilink flexible arms. ASME J. Dyn. Syst. Meas. Control 114:70-77 Vandergrift M W, Lewis F L, Zhu S Q 1994 Flexible-link robot arm control by a feedback linearization/singular perturbation approach. J. Robotic Syst. 11:591-603
Control of a 2-DOF manipulator with a flexible forearm 523 Wang D, Vidyasagar M 1991 Control of a class of manipulators with a single flexible link - part 1: Feedback lineafization. ASME J. Dyn. Syst. Meas. Control 113:655-661 Zhu S Q, Lewis F L, Hunt L 1994 Robust stabilization of the internal dynamics of flexible robots without measuring the velocity of the deflection. Proc. IEEE Conf Decision and Control, Orlando, pp 1811-1817

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