Department of Mechatronics Eng., Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran
This research focuses on proposing an optimal trajectory planning and control method of two link rigid-flexible manipulators (TLRFM) for Dynamic Object Manipulation (DOM) missions. For the first time, achievement of DOM task using a rotating one flexible link robot was taken into account in . The authors do not aim to contribute on either trajectory tracking or vibration control of the End-Effector (EE) of the manipulator; On the contrary, utilizing the powerful tool optimal control accomplishing a point-to-point task for TLRFM is the purpose of the current research. Towards this goal, the pseudospectral method will be developed to meet the optimality conditions subject to system dynamics and boundary conditions. The complicated optimal trajectory planning is formulated as a nonlinear programming problem and solved by SNOPT nonlinear solver. To make robust the response of optimal control against external disturbances as well as model parameter uncertainties, the control partitioning concept is employed. The controlled input is composed of an optimal control-based feedforward part and a PID-based feedback component. The obtained simulation results reveal the usefulness and robustness of the developed composite scheme, in DOM missions.
 H. H. Lee and Y. Liang, “A coupled-sliding-surface approach for the robust trajectory control of a horizontal two-link rigid/flexible robot”, International Journal of Control, Vol. 80(12), (2007), pp. 1880-1892.
 J. F. Peza-Solis, G. Silva-Navarro and R. Castro-Linares, “Control of a rigid-flexible two-link robot using passivity-based and strain-feedback approaches”, 7th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2010), Tuxtla Gutiérrez, Chiapas, México, (2010), pp. 476-481.
 A. Abe, “Trajectory planning for residual vibration suppression of a two-link rigid-flexible manipulator considering large deformation”, Mechanism and Machine Theory, Vol. 44, (2009), pp. 1627–1639.
 L. Z. Shi and M. B. Trabia, “Comparison of distributed PD-like and importance based fuzzy logic controllers for a two-link rigid-flexible manipulator”, Journal of Vibration and Control, Vol. 11, (2005), pp. 723–747.
 S. Akella, W. H. Huang, K. M. Lynch and M. T. Mason, “Sensorless parts orienting with a one-joint manipulator”, IEEE International Conference on Robotics and Automation, Vol. 3, (1997), pp. 2383-2390.
 M. Erdmann, “An exploration of nonprehensile two-palm manipulation”, International Journal of Robotics Research, Vol. 17(5), (1998), pp. 485-503.
 Y. Maeda, T. Nakamura and T. Arai, “Motion planning of robot fingertips for graspless manipulation”, Proceeding of IEEE International Conference on Robotics and Automation, (2004), pp. 2951 – 2956.
 Y. Maeda and T. Arai, “Planning of graspless manipulation by a multifingered robot hand”, Advanced Robotics, Vol. 19(5), (2005), pp. 501-521.
 . K. Miyazawa, Y. Maeda and T. Arai, “Planning of graspless manipulation based on rapidly-exploring random trees”, Proceeding of IEEE International Symposium on Assembly and Task Planning: From Nano to Macro Assembly and Manufacturing, (2005), pp. 7-12.
 A. Akbarimajd and M. N. Ahmadabadi, “Manipulation by juggling of planar polygonal objects using two 3-DOF manipulators”, Proceeding of IEEE/ASME International Conference of Advanced Intelligent Mechatronics, (2007), pp. 1-6.
 S. Akella, W. H. Huang, K. M. Lynch and M. T. Mason, “Planar manipulation on a conveyor with a one joint robot”, the 7th International Symposium on Robotics Research, (1995), pp. 265-277.
 K. M. Lynch and M. T. Mason, “Stable pushing: mechanics, controllability, and planning”, The International Journal of Robotic Researches, Vol. 15(6), (1996), pp. 533-556.
 K. M. Lynch and M. T. Mason, “Dynamic nonprehensile manipulation: controllability, planning, and experiments”, The International Journal of Robotics Research, Vol. 18(1), 1999,pp. 64-92.
 T. Tabata and Y. Aiyama, “Tossing manipulation by 1 degree of-freedom manipulator”, Proceeding of the IEEE/RSJ International Conference on Intelligent Robots and Systems, (2001), pp. 132−137.
 B. Beigzadeh, M. N. AhmadAbadi and A. Meghdari, “Two dimensional dynamic manipulation of a disc using two manipulators”, Proceeding of IEEE International Conference on Mechatronics and Automation, (2006), pp. 1191−1196.
 A. Akbarimajd, M. N.Ahmadabadi and B. Beigzadeh, “Dynamic object manipulation by an array of 1-DOF manipulators: Kinematic modeling and planning”, Robotics and Autonomous Systems Vol. 55(6), 2007, pp. 444–459.
 H. Miyashita, T. Yamawaki and M. Yashima, “Control for throwing manipulation by one joint robot”, Proceeding of IEEE International Conference on Robotics and Automation, (2009), pp. 1273-1278.
 K. Tahara, K. Maruta and M. Yamamoto, “External sensorless dynamic object manipulation by a dual soft-fingered robotic hand with torsional fingertip motion”, Proceeding of IEEE International Conference on Robotics and Automation, (2010), pp. 4309-4314.
 K. Tahara, S. Arimoto and M. Yoshida, “Dynamic object manipulation using a virtual frame by a triple soft-fingered robotic hand”, Proceeding of IEEE International Conference on Robotics and Automation, (2010), pp. 4322-4327.
 B. Tarvirdizadeh and A. Yousefi-Koma, “Dynamic object manipulation by a flexible robotic arm; theory and experiment”, International Journal of Robotics and Automation, Vol. 27(3), 2012, pp. 263-275.
 G. G. Rigatos, “Model-based and model-free control of flexible-link robots: A comparison between representative methods”, Applied Mathematical Modelling, Vol. 33(10), (2009), pp. 3906–3925.
 M. Vakil, R. Fotouhi and P. N. Nikiforuk, “Maneuver control of the multilink flexible manipulators”, International Journal of Non-Linear Mechanics, Vol. 44(8), (2009), pp. 831-844.
 D. Wang and M. Vidyasagar, “Transfer functions for a single flexible link”, Proceedings of IEEE International Conference on Robotics and Automation, Vol. 2, (1989), pp. 1042 - 1047
 X. Zhang, W. Xu and S. S. Nair, “Comparison of some modeling and control issues for a flexible two link manipulator”, ISA Transactions, Vol. 43, (2004), pp. 509–525.
 S. Choura and A. S. Yigit, “Control of a two-link rigid-flexible manipulator with a moving payload mass”, Journal of Sound and Vibration, Vol. 243(5), (2001), pp. 883-897.
 R. I. Milford and S. F. Asokanthan, “Configuration dependent eigenfrequencies for a two-link flexible manipulator: experimental verification”, Journal of Sound and Vibration, Vol. 222(2), (1999), pp. 191-207.
 A. V. Rao, D. A. Benson and G. T. Huntington, Algorithm 902: GPOPS, “A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using the Gauss Pseudospectral Method”, ACM Transactions on Mathematical Software, Vol. 37(2), (2010), pp. 22-38.
 P. E. Gill, W. Murray and M. A. Saunders, User’s Guide for SNOPT Version 7: Software for Large Scale Nonlinear Programming, 2006.
 D. A. Benson and G. T. Huntington, T. P. Thorvaldsen and A. V. Rao, “Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method”, Journal of Guidance, Control, and Dynamics, Vol. 29(6), (2006), pp. 1435-1440.
 D. Garg, M. A. Patterson, C. L. Darby, C. Francolin, G. T. Huntington, W. W. Hager and A. V. Rao, “Direct Trajectory Optimization and Costate Estimation of Finite-Horizon and Infinite-Horizon Optimal Control Problems Using a Radau Pseudospectral Method”, Computational Optimization and Applications, Vol. 49(2), (2011), pp. 335-358.
 D. A. Benson and G. T. Huntington, “A Unified Framework for the Numerical Solution of Optimal Control Problems Using Pseudospectral Methods”, Automatica, Vol. 46(11), (2010), pp. 1843-1851.
 D. Garg, M. A. Patterson, W. W. Hager, A. V. Rao, D. A. Benson and G. T. Huntington, “A Unified Framework for the Numerical Solution of Optimal Control Problems Using Pseudospectral Methods”, Automatica, Vol. 46(11), (2010), pp. 1843-1851.
 J. J. E. Slotine and W. Li, “Applied Nonlinear Control”, Prentice-Hall Inc., Englewood Cliffs, New Jersey, (1991).